A Single-Atom Heat Engine
Dissertation
Zur Erlangung des akademischen Grades
„Doktor der Naturwissenschaften“
am Fachbereich Physik, Mathematik und Informatik
der Johannes Gutenberg-Universität in Mainz
vorgelegt von
Johannes Roßnagel
geboren in Heidelberg
Mainz, Mai 2017

Abstract
Heat engines convert thermal energy into mechanical work, typically involving a large
number of atoms or molecules. In this thesis, I demonstrate that the working medium
of thermal machines can be reduced to the ultimate limit of one single atom. This is the
first experimental realization of a heat engine, where a single atom converts thermal
excitations into mechanical work. As the working agent I used a single 40Ca+ ion,
confined in a linear Paul trap with a special funnel-shaped geometry.
As a preparation for the experiments, I performed extensive numerical Monte-Carlo
simulations of our system with realistic parameters in order to demonstrate its feasibility,
to optimize the trap geometry, and to determine the expected range of the individual
parameters. In collaboration with the group of Prof. Dr. E. Lutz, we established an
analytical framework of the heat engine dynamics for a harmonic oscillator. We extended
this model as well as the simulations to non-thermal reservoirs to demonstrate the
corresponding capability for an increase of the engines performance.
For the experimental realization, I have driven the ion in a thermodynamic Stirling
cycle by coupling it alternately to a hot and a cold reservoir. Thereby, the ion generates
work from thermal excitations, which is extracted and stored in a designated degree
of freedom. I have mapped and fully characterized the thermodynamic cycles of the
engine for various temperature differences of the reservoirs by a direct and time resolved
measurement of the ion’s trajectory. However, this was only possible due to a novel
temperature measurement technique, that allows for determining temperatures in a
range between 0.1 and 100mK. This technique is based on the spectroscopy of narrow
dark two-photon resonances in the fluorescence spectrum of the ion and allows for time
resolved measurements of the ion’s thermal dynamics with a resolution of up to 5 µs.
From the measured thermodynamic cycles, I evaluated the output power P and the
efficiency η of the engine, which reach up to P = 3.4× 10−22 J/s and η = 0.28% at a
temperature difference of ∆T = 66(8)mK between the hot and the cold reservoir. These
results are consistent with the predictions from an analytical model. The ability not only
to generate work, but also to store it and to make it available to other oscillatory systems
is a crucial step towards the implementation of more complex thermodynamic systems.
The model character of the trapped ions as harmonic oscillators allows one to transfer
the findings from this thesis to other oscillatory systems. Furthermore, the versatility of
trapped ion systems will allow one to analyze fluctuations of the engines parameters,
which dominate on short timescales. With trapped ions initialized in the motional ground
state, future heat engines may operate at the boarder of the classical and the quantum
regime. Therefore, the presented work may inspire and initiate future thermodynamic
experiments and more fundamental engine systems.
III

Zusammenfassung
EineWärmekraftmaschine wandelt thermische Energie in mechanische Arbeit um, wobei
das Arbeitsgas in der Regel eine große Anzahl von Atomen oder Molekülen umfasst. In
der vorliegenden Arbeit zeige ich erstmals, dass das Arbeitsmedium einer Wärmekraft-
maschine bis zur Grenze eines einzelnen Atoms reduziert werden kann. Dabei verwende
ich ein einzelnes 40Ca+ Ion, welches in einer trichterförmigen Paul-Falle gefangen wird.
Zunächst habe ich umfassende numerische Monte-Carlo Simulationen durchgeführt,
die auf realistischen Parametern beruhen und die Realisierbarkeit des Experimentes
demonstrieren. Die Geometrie der Falle wurde auf Grundlage der Simulationen opti-
miert und der zu erwartende Bereich der einzelnen Parameter der Maschine bestimmt.
Zusammen mit der Arbeitsgruppe von Prof. Dr. Eric Lutz haben wir einen analytischen
Rahmen aufgestellt, in dem die Dynamik eines harmonischen Oszillators als Wärmekraft-
maschine beschrieben werden kann. Unser analytisches und numerisches Modell haben
wir auf nicht-thermische Bäder erweitert und gezeigt, dass sich so die Leistungsfähigkeit
der Maschine deutlich erhöhen lässt.
Um die Wärmekraftmaschine experimentell zu realisieren, habe ich ein einzelnes
gefangenes Ion abwechselnd an ein warmes und ein kaltes Bad gekoppelt, wodurch es
einen Stirling-Kreisprozess durchläuft. Das Ion wandelt thermische Energie in mechani-
sche Arbeit um, die in Bewegungsenergie gespeichert wird. Durch eine zeitaufgelöste
Messung der Bewegung konnte ich thermodynamische Kreisprozesse für verschiede-
ne Temperaturen der Bäder charakterisieren. Dies wurde erst durch eine neuartige
Temperatur-Messmethode möglich, bei der Dunkelresonanzen im Fluoreszenz-Spektrum
untersucht werden. Sie erlaubt eine Bestimmung der Temperatur des Ions im Bereich
zwischen 0.1mK und 100mK und ermöglicht erstmals eine zeitliche Auflösung von Ther-
malisierungsprozessen. Aus den Kreisprozessen konnte ich die Leistung P der Maschine
sowie deren Effizienz η ermitteln. Diese reichen bis zu P = 3.4 · 10−22 J/s und η = 0.28%,
und stimmen mit den Vorhersagen eines analytischen Modells überein. Die Möglichkeit
mechanische Arbeit nicht nur zu erzeugen, sondern auch zu speichern und für andere
Systeme nutzbar zu machen, ist ein wichtiger Schritt für eine Integration in komplexere
thermodynamische Systeme.
Durch die Modelleigenschaften von gefangenen Ionen als harmonische Oszillatoren
können die Erkenntnisse meiner Arbeit auf verschiedene Oszillator-Systeme übertragen
werden. In zukünftigen Experimenten soll die Maschine mit grundzustandsgekühlten
Ionen an der Grenze zwischen Quantenmechanik und klassischer Mechanik betrieben
werden. Das hoheMaß an Kontrolle ermöglicht eine Untersuchung der Fluktuationen der
gemessenen Parameter, die auf kurzen Zeitskalen dominieren. Die vorliegende Arbeit
wird die Entwicklung von zukünftigen Maschinen im Nanomaßstab inspirieren und
weitere Experimente mit thermodynamischen Systemen motivieren.
V

Contents
1. Introduction 1
1.1. From steam engines to single atoms . . . . . . . . . . . . . . . . . . . . . . 2
1.2. Experimental thermodynamics on the nanoscale . . . . . . . . . . . . . . . 4
1.3. Microscopic and nanoscopic heat engines . . . . . . . . . . . . . . . . . . . 5
1.4. Non-classical and quantum reservoirs . . . . . . . . . . . . . . . . . . . . . 7
1.5. Quantum thermodynamics and quantum heat engines . . . . . . . . . . . 8
1.6. Trapped ions as a model system . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7. Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2. General concept of the single-atom heat engine 13
2.1. The laws of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1. Zeroth law: thermal equilibrium . . . . . . . . . . . . . . . . . . . . 13
2.1.2. First law: energy conservation . . . . . . . . . . . . . . . . . . . . . 14
2.1.3. Second law: entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.4. Third law: absolute zero . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2. Thermodynamic cycles of heat engines . . . . . . . . . . . . . . . . . . . . . 15
2.2.1. The Otto cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2. The Stirling Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3. Power and efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1. Carnot efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2. Efficiency at maximum power . . . . . . . . . . . . . . . . . . . . . 18
2.4. A funnel-shaped ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1. The harmonic trapping potential . . . . . . . . . . . . . . . . . . . . 19
2.4.2. The tapered potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.3. Origin and reduction of micromotion . . . . . . . . . . . . . . . . . 21
2.5. Working cycle of the single-atom heat engine . . . . . . . . . . . . . . . . . 22
2.6. Laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7. The calcium ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3. Results of analytical and numerical investigations 27
3.1. The quantum harmonic oscillator as heat engine . . . . . . . . . . . . . . . 28
3.1.1. Thermal state of a quantum harmonic oscillator . . . . . . . . . . . 28
3.1.2. Otto cycle for a harmonic oscillator . . . . . . . . . . . . . . . . . . . 29
3.1.3. Work and efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2. Proof of concept - a single ion as working fluid . . . . . . . . . . . . . . . . 33
3.2.1. Concepts of numerical simulations . . . . . . . . . . . . . . . . . . . 34
3.2.2. Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . 35
VII
Contents
3.3. Employing non-thermal reservoirs . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1. Otto cycle with a squeezed thermal reservoir . . . . . . . . . . . . . 41
3.3.2. Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.3. Generation of squeezing . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.4. Remarks on the generation of work and the notion of efficiency . . 46
3.4. Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4. Setup of the experiment 49
4.1. The funnel-shaped ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1. The radio-frequency supply . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.2. The constant voltage supply . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.3. Electric field noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.4. Loading ions into the trap . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.5. Detection of fluorescence light . . . . . . . . . . . . . . . . . . . . . 55
4.1.6. Generation of a magnetic quantization field . . . . . . . . . . . . . . 58
4.2. The vacuum setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3. The laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1. Laser system for ionizing neutral Ca-atoms . . . . . . . . . . . . . . 59
4.3.2. Laser systems for cooling and detecting the ions . . . . . . . . . . . 60
4.3.3. Scanning and switching of the beams . . . . . . . . . . . . . . . . . 61
4.3.4. Frequency stabilization . . . . . . . . . . . . . . . . . . . . . . . . . 62
5. Coherent excitation of motion 65
5.1. Characterization of the ion trap . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.1. Measurement of trap frequencies . . . . . . . . . . . . . . . . . . . . 65
5.1.2. Trap frequencies in a tapered potential . . . . . . . . . . . . . . . . 68
5.2. Demonstration of the elements of the heat engine working principle . . . 69
5.3. Determination of the damping coefficient . . . . . . . . . . . . . . . . . . . 70
5.3.1. Step response to coherent drive . . . . . . . . . . . . . . . . . . . . . 70
5.3.2. Broadening of the resonance . . . . . . . . . . . . . . . . . . . . . . 74
5.4. Addressing of single atoms and controlled interaction . . . . . . . . . . . . 76
6. Incoherent thermal excitation 79
6.1. Temperature of a single trapped ion . . . . . . . . . . . . . . . . . . . . . . 80
6.1.1. The notion of temperature . . . . . . . . . . . . . . . . . . . . . . . . 80
6.1.2. Generation of thermal states at various temperatures . . . . . . . . 82
6.2. Principles of dark resonance thermometry . . . . . . . . . . . . . . . . . . . 85
6.2.1. Calculation of dark resonance spectra . . . . . . . . . . . . . . . . . 86
6.2.2. Dark resonances and thermal motion . . . . . . . . . . . . . . . . . 87
6.2.3. Shaping the laser cooling by exploiting dark resonances . . . . . . 90
6.3. Determination of the laser linewidth . . . . . . . . . . . . . . . . . . . . . . 91
6.4. Temperature measurements in thermal equilibrium . . . . . . . . . . . . . 93
6.4.1. Experimental parameters and characteristics . . . . . . . . . . . . . 93
6.4.2. Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.4.3. Measurement of steady-state temperatures . . . . . . . . . . . . . . 94
6.5. Investigation of thermal dynamics . . . . . . . . . . . . . . . . . . . . . . . 95
6.6. Engineering thermal states by exploiting dark resonances . . . . . . . . . . 96
6.7. Spatial thermometry technique . . . . . . . . . . . . . . . . . . . . . . . . . 98
VIII
Contents
6.8. Outlook and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7. Experimental implementation of the single-atom heat engine 101
7.1. Specific experimental parameters . . . . . . . . . . . . . . . . . . . . . . . . 102
7.2. Electronic control of the experiment . . . . . . . . . . . . . . . . . . . . . . 103
7.3. Resolved measurement of the ion’s oscillation . . . . . . . . . . . . . . . . 103
7.4. The thermodynamic cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.4.1. Measurement of the thermodynamic cycles . . . . . . . . . . . . . . 105
7.4.2. Analytical derivation of the thermodynamic cycles . . . . . . . . . 108
7.4.3. A Stirling-like engine type . . . . . . . . . . . . . . . . . . . . . . . . 110
7.5. Output power of the engine . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.5.1. Stationary output power . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.5.2. Start of the engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.6. Efficiency of the engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.7. Exclusion of non-thermal excitation of the ion . . . . . . . . . . . . . . . . . 118
7.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8. Future perspectives of the single-atom heat engine device 121
8.1. Single-atom heat pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.1.1. Single-ion operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.1.2. The three-ion heat pump . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.2. Autonomous, self-contained engine . . . . . . . . . . . . . . . . . . . . . . 124
8.3. Towards a quantum heat engine . . . . . . . . . . . . . . . . . . . . . . . . 125
8.3.1. A heat engine at the ground state of motion . . . . . . . . . . . . . . 126
8.3.2. The two-level system as heat engine . . . . . . . . . . . . . . . . . . 126
8.4. The tapered ion trap as a tool for quantum information . . . . . . . . . . . 127
Appendices 129
A. Appendix 129
A.1. Treatment of trap electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.2. Processing of PMT pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.3. Stroboscopic recording of the ions motion . . . . . . . . . . . . . . . . . . . 130
A.4. Radio-frequency signal generator for AOM driving . . . . . . . . . . . . . 131
A.5. Power stabilization during AOM scans . . . . . . . . . . . . . . . . . . . . 132
A.6. Switching between lasers with two VCOs . . . . . . . . . . . . . . . . . . . 133
A.7. Design of a next generation funnel-shaped ion trap . . . . . . . . . . . . . 135
B. List of publications 139
Bibliography 140
IX

List of Figures
1.1. Schematic diagram of a heat engine . . . . . . . . . . . . . . . . . . . . . . . 2
1.2. Comparison of different experimental heat engine systems . . . . . . . . . 6
1.3. Probability distribution of motional states in phase space . . . . . . . . . . 10
2.1. Thermodynamic cycles: the Otto and the Stirling cycle . . . . . . . . . . . 16
2.2. The linear Paul trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3. Geometry of the funnel-shaped ion trap . . . . . . . . . . . . . . . . . . . . 20
2.4. The linear and the tapered trapping potential . . . . . . . . . . . . . . . . . 21
2.5. Fundamental mechanism converting thermal energy into motion . . . . . 23
2.6. Working principle of the single ion heat engine . . . . . . . . . . . . . . . . 24
2.7. Mechanical analog of the single-atom heat engine . . . . . . . . . . . . . . 25
2.8. Energetic level scheme of 40Ca+ . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1. Otto cycle for a harmonic oscillator in the {n,ω}-representation . . . . . . 29
3.2. Analytic analysis of work and efficiency of the Otto cycle . . . . . . . . . . 32
3.3. Simulated energy-frequency diagram of the Otto cycle . . . . . . . . . . . 36
3.4. Simulated evolution of the radial energy during heat engine operation . . 37
3.5. Ensemble average of the ion trajectories during heat engine operation . . 37
3.6. Phase space distribution at different phases during an engine cycle . . . . 38
3.7. Efficiency of the Otto engine as a function of the temperature ratio . . . . . 39
3.8. Efficiency at maximum power . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.9. Simulated Otto cycle with a squeezed thermal reservoir . . . . . . . . . . . 44
3.10. Simulated efficiency at maximum power for different squeezing parameters 45
3.11. Widths of pure thermal and squeezed thermal states . . . . . . . . . . . . . 46
4.1. Images of the funnel-shaped ion trap. . . . . . . . . . . . . . . . . . . . . . 50
4.2. Exploded-view CAD-drawing of the ion trap setup . . . . . . . . . . . . . 51
4.3. Drawing of the helical resonator . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4. Radio frequency signal on the output lines of the helical resonator . . . . . 53
4.5. Electronic setup of the inversion circuit for electric noise . . . . . . . . . . 54
4.6. Recorded voltage noise signal . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.7. Camera image of a linear 5-ion crystal . . . . . . . . . . . . . . . . . . . . . 56
4.8. Acquisition process of the ICCD camera . . . . . . . . . . . . . . . . . . . . 57
4.9. Technical scheme of the laser setup . . . . . . . . . . . . . . . . . . . . . . . 60
4.10. Orientation of the individual laser beams with respect to the ion trap . . . 61
4.11. Setup of the laser frequency stabilization to a reference cavity . . . . . . . 62
XI
List of Figures
5.1. Amplitude of a harmonic oscillator as a function of the driving frequency 66
5.2. Measurement of the trap frequency by a modulated laser beam . . . . . . 67
5.3. Measured radial trap frequencies as a function of the axial position . . . . 68
5.4. Demonstration of the fundamental working mechanism of the heat engine 69
5.5. Deliberate dissipation and step response to a coherent drive . . . . . . . . 71
5.6. Determination of the damping coefficient . . . . . . . . . . . . . . . . . . . 72
5.7. Analysis of the damping coefficient by step response measurements . . . . 73
5.8. Broadening of the spectral resonances for different damping coefficients . 74
5.9. Characterization of the damping coefficient as a function of the parameters
of the dissipation laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.10. Addressability of single ions within a lon ion crystal . . . . . . . . . . . . . 77
5.11. Addressing of single ions in a 3-ion and a 4-ion crystal . . . . . . . . . . . 77
6.1. Spatial intensity distribution of a single ion at different temperatures . . . 82
6.2. Characterization of the voltage noise signal . . . . . . . . . . . . . . . . . . 83
6.3. Emergence of dark resonances due to coherent population trapping . . . . 86
6.4. Emergence of four dark resonances in the fluorescence spectrum of 40Ca+ 88
6.5. 3D-surface plot of the calculates thermal broadening of dark resonances . 89
6.6. Determination of the linewidth of the laser sources . . . . . . . . . . . . . . 92
6.7. Measured dark resonances at different temperatures . . . . . . . . . . . . . 94
6.8. Scan of a dark resonance in pulsed acquisition mode . . . . . . . . . . . . . 95
6.9. Time resolved thermal dynamics of trapped ions . . . . . . . . . . . . . . . 96
6.10. Influence of the dark resonance on the temperature of the ion . . . . . . . 97
6.11. Determination of the point spread function of the imaging optics . . . . . 98
6.12. Temperature measurement using the spatial thermometry technique . . . 99
7.1. Condensed sketch of the setup of the heat-engine experiment . . . . . . . 102
7.2. TTL circuit and timing of the heat-engine drive . . . . . . . . . . . . . . . . 103
7.3. Time resolved measurement of the ion’s oscillation . . . . . . . . . . . . . . 104
7.4. Temperature dynamics of the ion in contact with the thermal reservoirs . . 106
7.5. Simulated temperature dynamics of the heat engine . . . . . . . . . . . . . 107
7.6. Measured thermodynamic Stirling cycle . . . . . . . . . . . . . . . . . . . . 109
7.7. Output power of the engine . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.8. Start of the engine from rest . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.9. Efficiency of the engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.10. Exclusion of non-thermal excitation of the ion . . . . . . . . . . . . . . . . . 118
8.1. Thermodynamic cycle of a heat pump for a harmonic oscillator . . . . . . 122
8.2. Three ion heat pump configuration . . . . . . . . . . . . . . . . . . . . . . . 123
8.3. Segmented trap design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.4. Design for an autonomous operation of the single-ion heat engine . . . . . 125
8.5. Approaches to realize a quantum heat engine with a trapped-ion system . 126
8.6. Bow-tie trap for quantum computation . . . . . . . . . . . . . . . . . . . . 127
A.1. Electron-microscope images of the trap electrodes . . . . . . . . . . . . . . 130
A.2. Circuit diagram for stretching the single photon pulses from the PMT . . . 131
A.3. Stroboscopic record of the breathing mode of a two ion crystal . . . . . . . 131
A.4. Block diagram of the controllable rf-signal generator . . . . . . . . . . . . . 132
XII
List of Figures
A.5. Characteristics of the VCO module . . . . . . . . . . . . . . . . . . . . . . . 133
A.6. Step-response of an AOM as electro-optical shutter . . . . . . . . . . . . . 133
A.7. Stabilization of the defraction effciency of an AOM . . . . . . . . . . . . . . 134
A.8. Switching between two drive frequencies which for the AOM . . . . . . . 134
A.9. New design of the funnel-shaped ion trap . . . . . . . . . . . . . . . . . . . 135
A.10.Details and geometry of the new trap . . . . . . . . . . . . . . . . . . . . . 136
XIII

"What are the possibilities of small but movable machines? They may or
may not be useful, but they surely would be fun to make."
—Richard Feynman, 1959
1 Introduction
The laws of thermodynamics impact our daily life like almost no other physical law.
They describe the conservation of energy, its transformation from one form to another,
and govern the exchange of heat and the generation of work. Their obvious effects on
our everyday perception are as diverse as the process of cooking, the daily weather or the
means of transport. Our modern society is entirely based on the existence of machines
and engines, while the technological miniaturization has lead to the development of
increasingly smaller devices. One of the most intriguing questions of nanotechnology is
whether the concept of motors may be reduced to the nanoscopic level. Nature shows
us the way: muscles are outstanding molecular engines, with efficiencies far above
any machine created by humans so far. To devise nano engines, a thermodynamic
understanding of processes in small scale systems is essential. Statistical mechanics
allows one to reduce the thermodynamic behavior of large systems to processes on
the single particle level, and finally connects thermodynamics to quantum mechanics.
However, when treating single particles, the thermodynamic framework changes. While
classical systems are always in thermal contact with their surrounding, this interaction
has to be treated explicitly for nanoscopic systems, and a limited particle number leads
to fluctuating thermodynamic quantities. Even the physics of information processing
and computation has a thermodynamic contribution. For instance, information may be
converted into mechanical work and the erasure of information requires a minimum
amount of heat.
In the early days, the understanding of thermodynamics was based on phenomenolog-
ical findings. But by the time of the industrial revolution in 19th century it developed
into a scientific discipline. Since then, thermodynamics has spread into many fields of
research, such as engineering, biology, chemistry or nanotechnology. In this thesis we
implement a single-atom heat engine, which combines the engineering approach and
thermodynamics of nanoscopic systems. In order to introduce to these two disciplines,
we will sketch their historical evolution in the following. This introduction, however,
is intended to place the present work into its scientific context, but makes no claim to
historical completeness. Subsequently, we will summarize the state of the art of nano-
thermodynamic experiments and recent realizations of micro-sized heat engines. Finally,
we will give brief insights in the field of quantum thermodynamics, with a focus on a
variety of proposals for quantum heat engines, as well as on the technological progress
of ion traps and their role as model systems. We close the introduction with an outline of
this thesis.
1
1. Introduction
Figure 1.1. – A heat engine is a cyclically working device, that converts a difference in thermal
energy between a hot reservoir and a cold reservoir to some degree into mechanical energy.
Thereby, it converts non-directed, thermal motion of atoms into directed and coherent motion
or, in other words, heat into work.
1.1. From steam engines to single atoms
The early scientists and philosophers, from the era of Greeks and Romans to the industrial
revolution, tried to describe their daily perceptions on intuitive quantities, such as
temperature and pressure, and the relations between them. But with the invention of
thermal machines, there was a growing interest to understand the underlying processes.
The first industrial piston-based steam engine, invented by Newcomen in 1712 and
refined by Watt in 1765, gave rise a diversity of useful machines and engines [Ros10].
This encouraged scientists of the time to study the conversion of heat into work, in search
of devices with higher efficiencies, dreaming of perpetual motion machines. Based on
the conservation of thermal energy and the first ideas of entropy, called moment of activity
at the time, Carnot derived the maximum efficiency of any cyclically working machine
between two thermal reservoirs. He was the first to analyze the underlying physical
principles of heat engines [Car24], stating in 1824:
"The thermal agency by which mechanical effect may be obtained is the transference of
heat from one body to another at a lower temperature",
see Fig. 1.1. The understanding of heat being an individual form of energy was estab-
lished by von Mayer and Joule independently in 1842 and 1843. Joule formulated the
mechanical equivalent of heat [Rey92] as:
"Wherever mechanical force is expended, an exact equivalent of heat is always obtained",
an early expression of the first law of thermodynamics.
However, scientists realized soon that no real machine is able to reach the maximum
efficiency predicted by Carnot. The statements of Carnot and Joule failed to recognize
that transformations of energy in the real world don’t occur in isolated systems. The
second law is the logical consequence, taking the increase of entropy into account. It
breaks the temporal symmetry of the universe as it defines a specific direction of its
evolution in time. It was Clausius who formulated the second law in a modern form in
1850 [Cla50]:
"Heat can never pass from a colder to a warmer body without some other change, connected
therewith, occurring at the same time".
He also coined the term of entropy, based on Carnots moment of activity, and ends his
work "The mechanical Theory of Heat" [Cla67] with a famous summary of the first and the
second law:
"The energy of the universe is constant. The entropy of the universe tends to a maximum".
2
1.1. From steam engines to single atoms
The evolution of industrial engines took their next step in 1816, when Stirling invented
a closed cycle engine with external combustion, which was the first hot gas engine put to
practical work [Fin01]. As an alternative to the common steam engines of those times, the
Stirling engine is the prototype that most modern motors are based on. The simple setup,
the closed air cycle as well as a successful down scaling led to a revival of Stirling engines
in the mid of the 20th century as domestic motors, powered by a simple candle flame. The
most efficient hot air engine was created by Otto in 1876. As a further development of a
gas engine invented by Lenoir, this was the first internal combustion engine which used
a compression of the gas mixture prior to ignition in order to increase the efficiency. The
Otto engine started a veritable success story, which continues today with modern car
engines. Both hot air engine types find their continuation in modern science: while the
working principle of our single-atom engine builds on an Otto cycle, the experimental
realization resembles a Stirling cycle for technical reasons.
Themicroscopic interpretation of thermodynamics startedwith Bernoulli in 1738 [Ber68],
who described the origin of pressure as collisions of atoms with the walls. His ideas
were successfully pursued by Clausius [Cla57] and Maxwell [Max71]. In 1867, Maxwell
conceived a thought experiment to question the second law on the scale of single atoms.
He invented an intelligent being, later called demon, that is able to separate hot and cold
atoms in a closed system, and therefore decreases entropy [Max71]. Thereupon, Boltz-
mann tried to derive an analytical framework which reduces thermodynamic properties
entirely to the kinetic behavior of individual atoms [Cal85]. He put the entropy on a
statistical footing and derived its relation to the number of accessible microstates Ω with
his famous formula S = kB lnΩ. His work encouraged scientists of the time to inves-
tigate the consequences of microscopic processes on thermodynamic quantities. Since
at least 1905, when Einstein determined the Brownian motion to be a result of random
interactions with water molecules, the postulate that heat is a disordered movement of atoms
is an established result [Fli06]. Since then, stochastic thermodynamics has successfully
derived all macroscopic thermodynamic quantities from statistical properties of the
microscopic structure of matter [Sei12]. In 1961, Landauer fundamentally connected the
expanding field of information theory with thermodynamic processes. He stated that the
erasure of information is a dissipative process and requires a minimum amount of heat,
equal to the entropy of one bit of information with two microstates, QL = kB ln 2 [Lan61].
The Landauer principle states by implication that the entropy of a memory increases
when information is stored, and therefore ensures that any intelligent demon acts in
accordance with the second law.
The question of how to apply thermodynamics to single, isolated atoms arose with the
discovery of nucleation reactions in the 1930s [Vol26, Fes87]. In the past decades, there
is an ongoing size reduction of technical devices from the macroscale to the nanoscale,
which is nowadays approaching the ultimate limit of single atoms. This process of minia-
turization thus puts thermodynamics of single particles and nanosystems into the focus
of research [Cer09]. Amongst others, Jarzynski and Crooks propelled the investigation
of statistical processes on the nanoscale and for small particle numbers [Jar97, Cro99].
Their investigations demonstrate that there are substantial differences between ther-
modynamic processes for macrosystems and nanosystems. They showed that, as a
consequence of the statistical nature of single particle processes, the thermodynamic
quantities and state variables are subjected to fluctuations. These fluctuations can even
lead to statistical short term violations of the laws of thermodynamics, as predicted by
3
1. Introduction
Evans et al. [Eva93]. However, the laws hold, of course, for long time averages of the
systems dynamics [Jar11].
In his 1959 talk "There is plenty of room at the bottom", Feynman already envisioned
tiny motors working at the single-atom level [Fey60]. He challenged physicists of the
time to follow the visionary way of miniaturization, which had just begun, and to
make a start towards engineering machines at the ultimate limit of single atoms. The
start of the theoretical investigations of nanoscopic machines and devices is commonly
accredited to Hill and his work on "Thermodynamics of Small Systems" by 1963 [Hil63].
The pioneering work of Scovil and Schulz-DuBois on maser heat engines in 1959 [Sco59]
also triggered an intense and continuous effort in theoretical studies of heat engines
and refrigerators on the nanoscale including their quantum properties [Ali79, Kos84,
Gev92a, Lin03, Hen07]. In the 1970s and 1980s, Taniguchi and Drexler popularized the
potential of nanotechnology, which boosted the development of numerous technical
inventions on the nanoscale [Tan74, Dre86]. In the 1990s, a new series of molecular
synthesis techniques allowed for the creation of various mechanical machines based
mechanically linked molecules. On this basis, dramatic experimental progress in the
last decade has lead to the diverse micro- and nanosized heat engines, which will be
discussed in Sec. 1.3 in detail. The achievements in the miniaturization of machines, as
well as its potential for future nanotechnological devices, has been acknowledged with
the 2016 Nobel prize in chemistry for Sauvage, Stoddard and Feringa.
In this thesis, we follow a different approach and employ one single atom as working
agent of an engine, thus fulfilling Feynman’s dream [Roß16]. However, our engine was
not conceived to complete the process of miniaturization, but rather to inspire and to mo-
tivate subsequent experiments in the field of single-atom and quantum thermodynamics.
1.2. Experimental thermodynamics on the nanoscale
On the nanoscale, the classical framework of thermodynamics has to be expanded. The
classical description of the laws of thermodynamics has been formulated for mean values
of state variables in the thermodynamic limit, thus for ensembles with a large number
of particles on the order of 1023 [Cen15]. The progress in stochastic thermodynamics
has led to a transfer of the definitions of thermodynamic parameters to the level of
single trajectories of microscopic systems [Sei12, Alo16b]. The increasing level of control
of micro- and nanosystems in the past decade has allowed for a new generation of
thermodynamic experiments. The high precision in preparation and readout allows
to target former thought experiments, such as a Maxwell’s demon or the Landauer
principle.
A Maxwell’s demon could be realized recently in various nanosystems. In cold atom
systems a one-way barrier has been realized in search of effective cooling techniques,
which acts as a Maxwell’s demon sorting atoms according to their direction of mo-
tion [Tho08]. In the form of a Feynman ratchet, it permits the extraction of work from a
single heat bath. A spatial asymmetry in dissipation has been used to obtain a directed
motion from undirected Brownian motion and thus to perform work [Rou94, Par02]. In
photonic systems a Maxwell’s demon could be realized by electric feedback operations,
which sort photons of a thermal light source to extract work into an electric circuit [Vid16].
A similar approach has been pursued by Koski et al. with a single electron system, where
a Maxwell’s demon uses information about a thermal electron to generate work from
4
1.3. Microscopic and nanoscopic heat engines
its Brownian motion [Kos14a]. This process has even been inverted to use the demons
information to reduce the entropy of the system [Kos15].
The Maxwell’s demon experiments are closely linked to Landauers erasure theorem.
The relation between information and heat has been first demonstrated experimentally
in a classical system using a micro-particle in an optical dipole trap [Bér12]. Koski et al.
showed that the Landauer principle can be inverted and the amount of heat QL = kB ln 2
can be extracted from one bit of information [Kos14a]. Recently, the Landauer principle
has even been proven in the quantum regime employing a nuclear magnetic resonance
setup [Pet16].
The emerging variety of single particle systems has allowed for the investigation of the
statistical nature of the nano-thermodynamics in detail, as predicted by Jarzynski, Crooks
and Evans. Short term fluctuations of thermodynamic parameters have been experimen-
tally observed in different systems, as diverse as molecular systems [Col05, Ale12], levi-
tated nanospheres [Gie14], nuclear spin systems [Bat14] and single trapped ions [An15].
The fact that on short timescales these fluctuations can even lead to a violation of the laws
of thermodynamics has been demonstrated with nanoparticles [Wan02, Ver14a, Mar16].
However, the laws of thermodynamics hold, of course, for long time averages of the
systems [Jar11]. A profound review of the field of fluctuation theorems for small scale
thermodynamics has been composed by Seifert [Sei12], and more recently by Dinis et
al. [Din16].
The above examples illustrate the variety of possibilities which open up on the basis
of micro- and nano-sized system. However, most of the above cited experiments are still
in the classical regime of thermodynamics. If we decrease the temperature of the system
to the quantum regime, where the discreteness of energy levels becomes important,
we are able to test the laws of thermodynamics in the realm of quantum mechanics
experimentally. The use of quantum states instead of classical states for thermodynamical
experiments is an exciting technological challenge and will provide important insights
in the thermodynamics of quantum information processing. The increasing fidelity of
preparation, control and readout of quantum states promises a rise of thermodynamic
experiments in various quantum systems in the near future.
1.3. Microscopic and nanoscopic heat engines
The process of miniaturization has led to a growing interest in small scale engines, while
the technical achievements in increasing precision of modern experiments has provided
the necessary tools. The combination of both gave birth to several heat engines on the
micro- and nanoscale in the past couple of years, as summarized in the following and
sketched in Fig. 1.2.
Inspired by nature and by the biological motors in muscle cells, the creation of nano-
sized engines started with the synthesis of appropriate molecular structures. In 1994,
Sauvage et al. and Stoddard et al. both realized a mechanically moving device based on
linked molecules [Liv94, Bis94], today seen as the "embryo of a non-biological molec-
ular machine" [Fer16]. In 1999, Feringa et al. constructed the first notable molecular
motor, where two parts of a macro-molecule rotate against each other when exposed to
UV-light [Kou99]. This concept was refined and reduced to a single molecule in 2002
by Hugel et al. [Hug02]. They drove an optical transition between different structural
configurations of a single polymer molecule cyclically, whereby the molecule is length-
5
1. Introduction
a) c) e)
n 1030 1020 1010 105 102 10 1
b) d) f)
Figure 1.2. –Number of atoms n of the working agent of different heat engine systems. (a) Com-
mon car engine [car]. (b) Micro-sphere suspended in water and confined in an optical
tweezer [Bli12]. (c) Solid state engine system: a silicon beam drives a micro-mechanical
oscillator [Ste11]. (d) Two clouds of cold atoms at different temperature, connected by a
small channel [Bra13a]. (e) A single azobenzene molecule works in an opto-mechanical
cycle [Hug02]. (f) A single ion serves as working agent of the heat engine, this work [Roß16].
ened and contracted against an external force and thus performs work. This device, as
well as Feringas engine, demonstrates the opto-mechanical energy conversion and hence
are not heat engines in the proper sense. However, it demonstrated the possibility for a
deliberate generation of mechanical work on the nanoscale.
The conversion of thermal energy into electrical power was demonstrated in a micro-
electromechanical system in 2003 by Wahlen et al. [Wha03]. They used a small vapor
bubble between two piezoelectric membranes and converted the expansion and con-
traction of the bubble into voltage differences in the membranes. But with a bubble
size of 1000 µm in diameter and 60 µm high, this was still far away from targeting new
regimes of thermodynamics. This approach, however, was pursued by Steeneken et
al., who demonstrated in 2011 a fully functional micro heat engine with a solid state
working agent [Ste11]. They employed the resistive heating and thermal expansion of
a micro-sized silicon beam to drive a mechanical oscillator at resonance. Interestingly,
this engine runs with a continuous current and amplifies the oscillation of the load
autonomously due to the piezoresistive effect of the silicon beam.
A major milestone was reached in 2012 with the stochastic heat engine of Blickle and
Bechinger [Bli12]. They used a single melamine bead with 3 µm in diameter, suspended
in water and confined in the harmonic potential of an optical dipole trap. They analyzed
the Brownian motion of this particle while driving a Stirling cycle by a periodic variation
of the water temperature and the strength of the optical potential. Work and efficiency of
the engine are derived from the time dependent spread of the Brownian motion. Since
this system is working in the overdamped regime, both heat and work are dominated by
6
1.4. Non-classical and quantum reservoirs
fluctuations due to the stochastic nature of the interactions. This experiment is widely
acknowledged as the first micro-sized heat engine and has drawn much attention in the
community. Recently, this experiment has been refined by Martinez et al. [Mar16]: in
2016 they published an outstanding work, where they could demonstrate that stochastic
micro heat engines, although working in the classical regime, may exceed the Carnot
efficiency on short timescales with a finite probability, as a consequence of fluctuating
parameters. As mentioned above, on a long term basis the fluctuations balance out and
the system obeys the classical expectations.
A single-cycle engine, based on the thermoelectric effect, has been demonstrated with
cold atom clouds. Brantut et al. showed in 2013 that a directed atom current against
a chemical potential is generated during the exchange of heat between two clouds of
atoms at different temperature [Bra13a]. In 2015, an energy harvester has been realized
experimentally by Thierschmann et al. using a three terminal quantum dot, where
thermal fluctuations are directly converted into an electric current [Thi15].
All the above mentioned engine systems are working in the classical regime with
thermal heat baths and still employ a large number of atoms. In this thesis, we demon-
strate a heat engine reduced to a single atom, which represents the first thermal machine
operating at the single-atom level. We employ a trapped ion in a linear Paul trap as model
system. Trapped ions are a popular platform for experiments in the quantum regime,
such as for quantum information processing. Although our engine is operated at high
temperatures where a classical description is appropriate, it has the potential to enter a
temperature range where the quantization of energy levels has to be taken into account.
Likewise, the engine could be driven by or drive quantum states. Further specific propos-
als for future micro-sized heat engines have been published for different optomechanical
setups [Zha14a, Dec15], microwave cavities [Ber14], and quantum dots [Hum02], which
all have the potential to enter the quantum regime as well. However, an experimental
demonstration of a quantum heat engine is still lacking.
1.4. Non-classical and quantum reservoirs
A reservoir is ideally an infinite pool of energy in equilibrium and with constant state
variables, such as its temperature. When a system is in contact with a reservoir both
exchange energy and the systems state variables, which are free to change, converge
towards equilibrium with the reservoir. The heat capacity of the reservoirs is supposed
to be much larger than the one of the system, and thus the state of the reservoir is not
changed during interaction.
Standard heat engines run cyclically between two classical thermal reservoirs at dif-
ferent temperatures. The engine reaches thermal equilibrium alternately with both of
the two reservoirs, resulting in a thermal state at the reservoir temperature. But not all
engines operate in such a thermal equilibrium regime.
For macroscopic systems the reservoirs are treated by their mean values. For micro-
and nanoscopic systems, however, the substructure of the reservoirs has to be taken into
account and leads to fluctuating interactions. The constructive role of thermal fluctu-
ations is exploited, e.g., by molecular Brownian motors such as ratchets [Par02, Bro06,
Hän09, Sei12]. Also engines operating with nanospheres have shown such fluctuations,
as mentioned above [Mar16].
Furthermore, on the microscale there are diverse non-thermal energy sources available.
7
1. Introduction
There are, for instance, biological motors such as myosin proteins, which are based
on completely different mechanisms. They operate in contact with a single isothermal
reservoir and are driven far from equilibrium by external energy input, such mechanical,
photonic or chemical energy [Bro06, Hän09, Sei12]. Also synthetic molecular motors
are usually driven by electrochemical processes or concentration gradients. Recently,
a Stirling engine has been realized, operating between two bacterial reservoirs at the
same temperature but with different concentrations of bacterias, achieving high effi-
ciencies [Kri16]. Such engines are not bound anymore to limits which are based on
the assumption of thermal reservoirs, such as the classical Carnot limit [Scu03, Dil09].
Even the efficiency at maximum power can reach unity for strong driving [Sei11, Bro12].
However, the condition of positive entropy production still limits the thermodynamic
performance of non-thermal reservoirs.
For single trapped ions and other quantum systems, like cold atoms, nanomechanical
oscillators and optomechanical systems, there is variety a different artificial and non-
thermal reservoirs available [Poy96, Lüt98, Rab04, Tom12, Woo14]. In this context, it
is convenient to regard the non-thermal reservoirs as perturbed thermal reservoirs,
characterized by their temperature and additional properties, which characterize their
deviation from equilibrium [Aba14]. Theoretical studies have indicated that the efficiency
of an enginemay be increased beyond the classical limits by coupling it to engineered non-
classical or nonequilibrium reservoirs, such as quantum coherent [Scu03] or quantum
correlated [Dil09] reservoirs. Recent experimental advances in reservoir engineering
techniques provide the necessary tools to realize a bath-like environment for a trapped
ion with a relaxation towards a non-thermal state [Mya00, Tur00b, Wan13b, Kie15].
1.5. Quantum thermodynamics and quantum heat
engines
For small thermodynamic systems at the nanoscale and at low temperatures, quantum
effects become important and have to be fully taken into account, for the systems as well
as for the reservoirs [Mil16]. Consequently, the reservoirs provide quantum properties
and transfer them to the system, if an appropriate coupling between both is established.
The emerging field of quantum thermodynamics is, on the one hand, an important
notion for all quantum systems and the basis to understand their thermal interactions,
such as equilibration and thermalization [Kos13, Vin15]. On the other hand, quan-
tum systems can be used to perform thermodynamic experiments in order to study,
e.g., quantum phase transitions or thermal energy transport [Ber13, Eis15]. Recently,
tremendous progress was achieved in engineering reservoir interactions for quantum
states [Mya00, Tur00b, Wan13b, Kie15]. However, the notion of thermodynamic limita-
tions has to be altered for quantum systems. In certain settings the second law splits
into a whole family of second laws in order to meet the various possible transforma-
tions [Bra15a].
If we push thermodynamics towards the quantum regime, new questions arise imme-
diately: what impact do quantum effects have on the performance of heat engines? In
the last few decades, an intense theoretical effort has been devoted to the investigation
of the potential and the limitations of engines in the quantum regime, e.g. for spin
systems [Gev92b] or for three level systems [Ali79, Gev96]. More concrete proposals for
8
1.6. Trapped ions as a model system
quantum engines employ cold atoms [Fia12] , opto-mechanical devices [Zha14a] or elec-
tronic quantum circuits [Esp10, Pek15]. In particular, theoretical studies have indicated
that quantum engines may outperform their classical analogs and provide efficiencies
beyond the limits of classical engines, thereby employing engineered quantum coher-
ences [Scu03, Scu10, Hor13, Dor13, Los15], as well as entanglement and quantum correla-
tions [Dil09, Hov13, Gal14] to fuel the engine. Specifically, the aspect of quantum refriger-
ators for cooling quantum systems efficiently has attracted great interest [Lin10b, Lev12b].
The existence of quantized energy states can be exploited to tailor the energetics of the
cooling mechanisms [Kol12, Lev12a]. Even hot energy sources can be used to cool
quantum system, similar to an absorption refrigerator [Mar12, Cle12, Lev12b]. Further-
more, the thermodynamic aspect of information processing becomes important for the
operation of small-scale quantum engines [Par15, Goo16].
Despite the multitude of proposals and theoretical investigations, there is no experi-
mental demonstration of a quantum heat engine. This fact is mainly due to the limitations
of the individual experimental systems, or on account of technical difficulties in reaching
the quantum regime. Recent progress in the degree of control has enabled first steps
being taken now [Thi15, Bra13a], but still it is unclear what quantum effects should be
probed. However, at the quantum limit we will meet severe challenges: a key issue
is the reservoir interactions with the quantum system. On the one hand there is a hot
bath, which should not heat the system beyond the limit of resolved quantum states. On
the other hand there is a cold bath, whose temperature is strictly limited by the third
law [Gem04]. These tight limits can be circumvented when using quantum states other
than thermal states and neatly engineered reservoirs. However, the interaction of quan-
tum states with any form of environment is challenging, since those states are extremely
sensitive to disturbances and fluctuations and tend to loose their coherence [Fri15].
Preparing the way towards the first experimental realization of a heat engine which
operates in the quantum regime and exploits non-classical energy sources is one of the
major motivations of this thesis. With a single trapped ion we have a system at hand
which is intrinsically a quantum system and which has a model character for other
harmonic oscillator systems. Therefore, it represents a good starting point for further
study of quantum engines.
1.6. Trapped ions as a model system
The trapping potential in a common Paul trap provides a quadrupolar spatial shape.
Hence the motion of a trapped ion can be described by a harmonic oscillator to very good
approximation [Lei03]. Therefore, trapped ions provide an ideal and general model for
other oscillatory system, such as micro- and nano-mechanical oscillators [Kar09, O’C10,
Lin10a, Fau12], offering a broad spectrum of potential applications. Further, trapped
ions provide an exceptional degree of control in their preparation and manipulation,
great precision in the measurement of their parameters [Lei03], and the capability of
coupling to engineered reservoirs [Poy96, Mya00, Kie15]. They permit cooling to the
ground state of motion and versatile state preparation by interaction with laser radiation.
For these reasons, they have played a prominent role in the experimental study of quan-
tum simulations [Bla12], as well as quantum computation and information processing
applications [Häf08, Bla08, Mon13]. In the past years, they also became invaluable tools
for the investigation of quantum thermodynamics [Haz12, Ulm13, Pyk13, Ram14, An15].
9
1. Introduction
Figure 1.3. – Probability distribution of the relevant motional states of a single trapped ion in phase
space (momentum – position) representation. (a) Cold thermal state of an ion at rest. The
probability is distributed normally and centered at {x = 0, p = 0}. (b) Hot thermal state.
The extent of the normal distribution increases with increasing temperature. (c) Coherent
state. The distribution of the cold ion is displaced from the origin and rotates with elapsing
time around it. In position space this corresponds to a sinusoidal oscillation. (d) Squeezed
thermal state. The probability distribution is asymmetric and can be decomposed in two
normal distributions with different widths along the two main axes in phase space. The state
rotates with elapsing time, which corresponds to a breathing motion in position space.
The harmonic oscillator for a quantum system exhibits different states of motion,
which represent the basis of our engine dynamics. Fig. 1.3 presents and compares the
characteristics of different states of motion of the harmonic oscillator in phase space,
which are relevant for the investigations throughout the present thesis: thermal states,
coherent states and squeezed thermal states. When in contact with a thermal reservoir,
the oscillator equilibrates towards a thermal state. It provides a thermal Boltzmann
distribution for the individual phonon states, leading to a normal probability distribution
in space and does not show any coherences or correlations. With higher temperature, the
spatial extent increases. A coherent state corresponds to a harmonic oscillation of the ion
in trapping potential and most closely resembles to a classical oscillation. A squeezed
state provides a non-uniform distribution of its phase space quadratures, ∆x 6= ∆p.
A squeezed thermal state is a thermal state whose circular phase-space distribution is
squeezed to some degree to an elliptical shape with the same area.
1.7. Outline of the Thesis
This thesis is organized as follows: first, in Ch. 2, we give a short introduction to the
general concept of the single-atom heat engine. We explain the fundamental coupling
mechanism of individual degrees of freedom, which lead to the conversion of thermal
energy into mechanical work. The fundamental considerations are accomplished by
analytical and numerical investigations of the proposed engine system in Ch. 3. These
studies demonstrate the feasibility of our engine system and define framework con-
ditions for the following experiments. Thereby, we show how the engine may run at
maximum output power and how non-thermal properties may be exploited to increase
the conversion efficiency between input energy and output power. The experimental part
starts with Ch. 4 and a detailed description of the experimental setup. This is followed
by a characterization of the coherent motion and the effect of thermal excitation of the
single trapped ion in Ch. 5 and Ch. 6, respectively. The first comprises a characterization
of the ion trap and an analysis of the artificial dissipation in our system, while the latter
treats heating and cooling processes of the ion, as well as the description of a specially
developed thermometry technique. Finally, in Ch. 7, we present the realization of a
10
1.7. Outline of the Thesis
single-atom heat engine. We derive the working cycles of the engine from measured
data and characterize the output power as well as the efficiency as a function of the
temperatures of the individual heat baths. We end this thesis with Ch. 8 and an overview
of future perspectives and possible applications of our engine system.
11

2 General concept of the single-atom
heat engine
This chapter gives a general overview of the fundamental principles of the single-ion
heat engine. We first take a closer look at the laws of thermodynamics, focusing on their
implications on the single-atom heat engine. In the following we present two of the
most prominent types of engine cycles: the Otto cycle and the Stirling cycle. Both will be
used to characterize our numerical and experimental findings in the course of this thesis.
We introduce relevant characteristics of heat engines, namely power and efficiency, and
discuss their limitations. In a second part we focus on the trapped ion system and present
the central element of our engine, the funnel-shaped trapping potential. With this at
hand, we give a short overview of the concrete working procedure of the single-ion heat
engine. Finally, we end this chapter with a discussion of the reduced level scheme of Ca+
ions, which we use as working agent in our engine.
2.1. The laws of thermodynamics
The four laws of thermodynamics provide a framework, within which thermodynamic
processes take place. They are defined as axioms, based on phenomenological findings.
Their formulation uses terms like system and environment, where the latter usually acts
as a thermal reservoir to the system. However, in realistic situations the dividing-lines
between system, environment and the surrounding universe are blurred. In general,
the reservoirs are assumed to be large enough such that the thermodynamic laws are
fulfilled, but strictly speaking the entire universe accounts for thermodynamic balances.
The framework of thermodynamics is a self-consistent and independent physical theory,
only based on the assumptions of the following four laws.
2.1.1. Zeroth law: thermal equilibrium
If two thermodynamic systems A and B are each in thermal equilibrium with a third
system C, then A and B are in thermal equilibrium with each other [Max71]. This equiv-
alence relation implies the foundation of the universal temperature T and does explicitly
not depend on the specifications of the bodies A, B and C or any other parameters. The
temperature is defined as the intensive variable of state which is equal for two systems,
if they are in thermal equilibrium [Cal85, Atk10].
13
2. General concept of the single-atom heat engine
2.1.2. First law: energy conservation
Energy can only be transformed from one form to another form, it cannot be created
or destroyed. The internal energy U is the extensive variable of state which accounts
for the total energy available for thermodynamic transformations. In this context, the
first law says that the internal energy is changed by the balance of heat Q and workW
contributions,
∆U = δQ+ δW, (2.1)
and is constant for any isolated system. This law establishes Q andW as different, but
equivalent modes of energy transfer, and implies the conversion of one into the other.
Note, that Q and W are no state variables but describe a flow of energy, they denote
processes not objects. The added heat δQ = m c ∆T is the amount of thermal energy that
changes the temperature of a body with mass m and heat capacity c by ∆T. The work
δW = −F ∆s is the mechanical energy input that moves a body by ∆s against an external
force F. The conversion of one into the other is the basis of all thermal machines.
Furthermore, the conservation of energy implies the non-existence of perpetual motion
machines, which generate more energy than they consume.
2.1.3. Second law: entropy
Clausius expressed the second law in terms of a defined direction of heat flow [Cla67],
saying that
"Heat never passes spontaneously from a cold to a hot reservoir, without work being done
on the system".
Thereby, this statement introduces irreversibility as a fundamental asymmetry in nature.
The state variable entropy S is changed by a heat flow and formally expressed in the
Clausius inequality,
≥ δQ∆S , (2.2)
T
where the equality applies to fictive reversible processes. Including the entropy, the
Clausius statement says that all spontaneous processes in isolated systems lead to an
increase of its entropy.
Regarding the operation of thermal machines, this statement claims that the heat sink
is as important as the heat source. According to Eq. (2.2), the entropy of the heat source
is reduced when heat flows out. As a consequence, a cold reservoir is required, where a
part of the heat must flow in to increase the entropy to at least the same amount. This
finding has led to the formulation of the second law by Kelvin and Planck [Wer66]:
"It is impossible to devise a cyclically operating device, the sole effect of which is to absorb
energy in the form of heat from a single thermal reservoir and to deliver an equivalent
amount of work".
Thus the positivity of entropy production limits the performance of heat engines, as it
is impossible for any engine to convert heat into work entirely. The limits of validity of
the second law of thermodynamics, especially in terms of information theory, have been
pointed out by Maxwell [Max71], Szilard [Szi29], Landauer [Lan61] and others early on,
but are subject to intensive discussions until today [Par15, Goo16].
Note, that the given definitions apply to purely thermal, classical and macroscopic
systems, based on state variables such as temperature, energy or entropy, which are
defined as mean values of large atomic ensembles.
14
2.2. Thermodynamic cycles of heat engines
2.1.4. Third law: absolute zero
The temperature provides a lower limit, its absolute zero T → 0K, a state with the
minimum possible energy. However, this state is not attainable by a finite number of
cyclic processes, as recognized by Nernst in 1905 [Ner69]. The third law defines an
absolute reference for the temperature scale as well as for the entropy, since the absolute
zero of temperature equals the zero point of entropy, S → 0. To be more precise, for
quantum systems with degenerate ground states with multiplicity q, the lowest entropy
equals S0 = kB ln q.
For more than a century the quest for absolute zero fascinated many researchers.
In recent years this topic has seen a revival in the context of quantum thermodynam-
ics [Rez09, Rai11, Kol12, Tor13]. Indeed, having temperatures of physical systems ap-
proaching absolute zero is crucial to modern quantum experiments. To identify the
optimal cooling rates close to absolute zero, quantum machines and, in particular, quan-
tum refrigerators play an important role [Lev12a].
2.2. Thermodynamic cycles of heat engines
Heat engines are cyclic operating systems, which convert a temperature difference
between two thermal reservoirs to some amount into mechanical work. Thereby a
working agent passes a thermodynamic cycle, where thermalization with reservoirs
and free evolution processes alternate. During such cycles, the agent exchanges heat
and work with the environment. The system acts as a heat engine if the work output
dominates the work consumption, powered by a net heat absorption. It acts as a heat
pump if a net amount of work is consumed in order to generate a heat output larger than
the heat input.
The internal energy of a system is expressed as a product of a conjugate pair of
an intensive and an extensive state variable, like for a classical macroscopic system
pressure and volume (mechanical quantities) or temperature and entropy (thermal quan-
tities) [Cal85, Atk10]. Using such pairs for a representation of a thermodynamic cycle
leads to a clear formal description, where the area of the cycle equals the net mechanical
work produced. From the relevant pairs of conjugate variables, all thermodynamic
properties can be derived.
In view of the large number of different combinations of thermodynamic processes,
we can define different idealized cycles, usually named after their inventor. For a clear
description of the idealized processes, the cycles usually are defined such that one state
variable is kept constant during this process. For real engines, the thermodynamic cycles
may deviate significantly from their idealized counterparts. In the course of this thesis
the Otto cycle and the Stirling cycle will play a major role.
2.2.1. The Otto cycle
The idealized Otto cycle consists of four individual processes [Cen15]. Heating and
cooling processes are much faster than the dynamics of the piston and the system is
isolated from the environment during the free evolution processes. These assumptions
lead to a clear separation of the individual strokes. The Otto cycle describes most spark-
ignition internal-combustion engines, such as car engines. It consists of two isochoric
temperature changes, where heat is transferred between the agent and the reservoirs at
15
2. General concept of the single-atom heat engine
Otto cycle Stirling cycle
2
2
1
1
3
4 3
4
Entropy Entropy
Figure 2.1. – Temperature – entropy diagrams for thermodynamic cycles. Otto cycle: (1) The hot
reservoir (red shaded) supplies heat to the the working agent, temperature and entropy
increase at constant volume. (2) The systems is isolated from the environment and expands
freely at constant entropy, increasing its volume. The engine performs work. (3) The remaining
heat is dumped in the cold reservoir (blue shaded) at constant volume, temperature and
entropy decrease. (4) The working agent is compressed at constant entropy, work is performed
on the system. Stirling cycle: (1) see Otto cycle. (2) The system expands while in thermal
equilibrium with the hot bath, increasing volume and entropy. The engine performs work.
(3) see Otto cycle. (4) The system is compressed while in contact with the cold bath, work
is performed on the system. The area of the cycles correspond to the net amount of work
produced during each cycle.
constant volume, and two isentropic processes, where the system is isolated from the
environment and changes its volume at constant entropy. The latter are supposed to be
adiabatic and reversible processes. During the expansion phase the agent moves a piston
and thus performs work, while during the compression phase works is consumed to
compress the agent by a displacement of the piston. Since the expansion occurs at higher
pressure and temperature than the compression, the amount of generated work exceeds
the work consumed. For details see Fig. 2.1 (a).
A main characteristic of the Otto cycle is the clear separation of defined reservoir
interactions and free evolution processes, which applies well to the situation of a single
trapped ion. The ion is suspended in vacuum and generally isolated from external
influences, allowing for adiabatic processes. Thermal interactions occur deliberately
through controlled interactions with laser beams and electric field noise. Therefore,
theoretical and analytical investigations of our system concentrate on the Otto cycle.
2.2.2. The Stirling Cycle
The experimental realization of the single-atom heat engine will be described by a
Stirling cycle [Fin01]. The Stirling engine is an idealized model for external combustion
engines. As compared the Otto cycle, in a Stirling cycle the working agent is coupled
permanently to one of the heat baths. Therefore, the isentropic processes are replaced
by isothermal processes. During the expansion stroke, heat is absorbed and work is
generated simultaneously. The compression stroke is the inverted process, where the
agent consumes work and dispenses heat. Fig. 2.1 compares the Stirling cycle to the Otto
cycle. As we will show in Sec. 7.4, the ion will be permanently coupled to one of the heat
baths for technical reasons, and thus a description by a Stirling cycle is appropriate.
16
Temperature
Temperature
2.3. Power and efficiency
2.3. Power and efficiency
For a practical application of heat engines, the key characteristics are power and efficiency.
Power is the net work of the engine per cycle, divided by the duration for one revolution,
P = W/τc. The work is the amount of mechanically usable energy, which is able to drive
a mechanical load. The ability of extracting the generated work from the engine system
and transferring it to a load is a crucial issue for nano-sized heat engines. In contrast to
most of the nano-engine systems, we will show that our single-ion setup provides the
intrinsic feature to transfer the work produced to a separate mode of oscillation and to
store the energy in its amplitude. However, in this thesis we demonstrate the working
principles of a single-ion engine without using the stored work in a mechanical sense,
but dissipating it in order to measure and to characterize it.
The efficiency of an engine is the relative amount of thermal energy which is absorbed
from the hot reservoir and converted into mechanical work, η = −W/Qh; the remaining
heat is dumped in the cold reservoir. Note, in this respect, that positive signs for heat and
work imply an energy flow from the environment to the engine system, while negative
signs signify that energy flows from the system to the environment.
One of the basic and important assumptions in thermodynamic balances is that the
reservoirs are given and the energetic costs for preparing them are not taken into ac-
count [Cal85, Cen15]. If these costs would be included, for instance in the calculation of
efficiencies of heat engines, the energy conservation in the entire universe would lead
to a vanishing efficiency for any engine: a reservoir can’t provide more energy than it
costs to prepare the reservoir. This is similar for car engines, where the energetic costs
for generation, pumping and refinement of oil are not taken into account.
2.3.1. Carnot efficiency
Sadi Carnot showed already in 1824 that the efficiency of any cyclic operating engine is
limited and that the maximum efficiency solely depends on the temperatures of the hot
and the cold reservoir, not on working fluid nor the characteristics of the cycle [Car24]:
"The motive power of heat is independent of the agents employed to realize it; its quantity
is fixed solely by the temperatures of the bodies between which is effected, finally, the
transfer of caloric1"
The Carnot principle is a direct result from the second law of thermodynamics, ∆S ≥
Q/T. If we neglect internal losses, the generated work is the difference between the heat
absorbed and the heat emitted, −W = Qh +Qc, and the efficiency reads
−W Qh +Qcη = = . (2.3)
Qh Qh
After one passage of the cyclic process, the total entropy of the environment, namely of
the two reservoirs, must increase, which restricts the amount of heat exchanged with the
reservoirs,
∮
Q Q
dS S S − h − c ≥ 0 ⇒ TQ ≤ −Q c= ∆ h + ∆ c = c h , (2.4)Th Tc Th
1 Nowadays, we call the motive power of heat the maximum efficiency of a cyclic and reversible engine, and the
notion of caloric corresponds to the exchange of heat.
17
2. General concept of the single-atom heat engine
where the equal sign corresponds to reversible processes. Inserting Eq. (2.4) in Eq. (2.3),
we obtain the range of possible efficiencies
Tc
η ≤ 1− , (2.5)
Th
where the Carnot efficiency ηC = 1− Tc/Th represents the upper bound.
2.3.2. Efficiency at maximum power
For theoretical investigations, ηC is an interesting characteristic of any engine system.
However, for the Carnot cycle the tota∮l entropy of the system and its environment
remains constant after one revolution, dS = 0. The cycle then has to be an entirely
reversible and quasistatic process. This means that the heat baths have to be infinitely
large and the thermalization processes infinitely slow, resulting in a vanishing power
output of the engine. For real engines, the thermalization processes are irreversible and
the cycles are optimized for maximum power output. As a consequence, to maximize
the power of an engine, efficiency is sacrificed; the power, not the efficiency, is optimized
for practical purposes.
The endoreversible model is a minimal extension of the Carnot cycle where the in-
ternal dynamics of the engine are supposed to be reversible but the heat transfer is
irreversible [DV95]. Formally, we introduce a thermal conductor which is in contact
with the hot bath at Th. Due to its finite heat capacity g, the other end of the conductor,
which is in contact with the engine, provides a lower temperature T′h < Th. The heat flux
through the conductor then is
Qg = g(T
′
h − Th). (2.6)
This heat is provided to the engine,Qg = Qh. The engine convertsQh into work generates
a work output according to Eq. (2.3) and using Eq. (2.5)
( )
T ( )−W = ηQh = g 1− c′ Th − T′T h . (2.7)h
We optimize the output power with respect to the variable temperature T′h and find the
optimality condition
dW √
= 0 ⇒ T′ = T T . (2.8)
dT′ h h ch
Supposing a reversible engine, η = 1 − T ′c/Th, the condition Eq. (2.8) results in the
efficiency at maximum power [Lef87]
√
Tc
η∗ = 1− , (2.9)
Th
√ √
with a maximum work output of −W 2max = g( Th − Tc) . The expression Eq. ( 2.9) is
commonly attributed to Curzon and Ahlborn, who derived it in 1975 [Cur75], although
it has been calculated by Yvon [Yvo55] , Novikov[Nov58] and Chambadal [Cha58]
independently twenty years before. However, throughout this thesis we will follow the
majority and refer Eq. (2.9) to as Curzon-Ahlborn efficiency. It is interesting to note that
this efficiency relation is independent of the kind of thermodynamic cycle or the nature
of the working agent, but only depends on the temperatures of the heat baths.
18
2.4. A funnel-shaped ion trap
a) b) c)
RF DC
x y DC RF
z x,y
Figure 2.2. – The linear Paul trap. (a) Geometry comprising two endcaps and four elongated rod-
electrodes. The endcaps (yellow) are grounded or biased with DC voltages. Usually, two
opposing radial electrodes (red) provide an oscillating voltage, while the other two electrodes
(blue) are grounded. (b) Electric field lines in the (x,y)-plane. The oscillating quadrupole
field is confining the ion in the center of the trap. (c) The mean potential, arising from the
oscillating fields, provides a harmonic confinement in the (x,y)-plane. The ion oscillates with
x,y(t) ∝ sin(ωx,yt), where the trap frequencies ωx,y depends on the strength of the trapping
potential.
2.4. A funnel-shaped ion trap
2.4.1. The harmonic trapping potential
In Paul traps the ions are spatially confined by a combination of static and oscillating
electric fields [Pau90, Lei03]. The most widespread kind of an ion trap is a linear Paul
trap, where the trapping potential is rotationally symmetric and the equipotential planes
provide ellipsoidal shapes. A simple design of a linear Paul trap is shown in Fig. 2.2 (a).
It consists of two endcaps and four elongated electrodes in a parallel and symmetrical
arrangement. To fulfill the Laplace equation ∆Φ = 0 for the trapping potential Φ inside
the trap, a combination of static and oscillating electric fields is mandatory [Lei03]. A
constant voltage is applied to the endcaps and provides a static confinement in z-direction
(hereinafter referred to as axial). In the x,y-plane (referred to as radial) the symmetrical
arrangement of the electrodes leads to an electric quadrupole field, see Fig. 2.2 (b).
Applying an oscillating voltage with an amplitude Urf at a radio-frequency Ωrf to two
opposing electrodes leads to an oscillating saddle potential
U
V (t) rf∝ 2 2rf r2
sin(Ωrf t) · (x − y ), (2.10)
0
where r0 is the distance of the electrodes to the trap axis. Due to the parabolic shape of the
potential, the focusing force which drives the ions towards the minimum of the potential
is always stronger than the defocusing force, similar to the strong focusing effect [Cou53].
Averaging over the fast oscillations of Vrf(t) leads to a linear ponderomotive restoring
force
d2~x(t)
~F(t) = m 2 = −α~x(t), (2.11)dt
proportional to the displacement ~x from the coordinate origin and where m is the mass of
the particle and k a constant relative to the potential strength [Lei03]. As a consequence
of ~F = −∇~ Φ, the ion behaves like being trapped in a static harmonic potential in all
19
Potential
2. General concept of the single-atom heat engine
x y
z
Figure 2.3. – Geometry of the funnel-shaped Paul trap. It provides an angle θ between the rod
electrodes and the trap axis z. To avoid massive axial micromotion resulting from the tapered
geometry, we apply the same oscillating voltage signal to each pair of opposing electrodes (red
and orange) but with inverted sign, see text. The endcaps (yellow) are biased with individual
DC voltages.
three dimensions
1 ( )
Φ 2p = αxx + αyy
2 + α z2z , (2.12)2
see Fig. 2.2 (c). Since Φp is not a solution of the Laplace equation, it is often referred to as
pseudo-potential. The solution of the equation of motion,
d2~x(t)
m 2 = −∇~ Φp, (2.13)dt
is a three dimensional harmonic oscillation xi(t) = ∑i Ai cos(ωit+ φi), with independent
oscillating frequencies ω2i = αi/m in all dimensions i ∈ {x,y,z}. If we introduce artificial
friction to the system, the equation of motion is expanded by a term which takes into
account the velocity dependent loss of energy. Additionally the system may be driven by
an external and time dependent force Fd(t), resulting in a inhomogeneous differential
equation,
d2x dx
2 + 2γ +ω
2x = Fd(t), (2.14)dt dt
with damping coefficient γ. For a weakly damped harmonic oscillator (γ ≪ ωz) the
dynamics of the ion are described by a modified oscillation frequency ω2d = ω
2 − γ2.
2.4.2. The tapered potential
In this work we employ a specially designed linear Paul trap with a tapered geometry.
In this design, the radial trap electrodes provide an angle θ with respect to the trap axis
z, resulting in a funnel-shaped geometry which is sketched in Fig. 2.3. While in axial
direction the static confinement stays unchanged, we reach a dynamic confinement of
the ion in the radial directions through a modified oscillating parabolic saddle potential
of the form
U
V (x,y,z,t) rf
sin(Ωrf t) U
rf ∝ 2 · (x
2 − y2) + dc · z2. (2.15)
(r0 + z tan θ) z20
20
2.4. A funnel-shaped ion trap
x
a)
z
x y
z
x
b)
z
Figure 2.4. – Cross sections of the trapping potential (x,z)-plane. (a) Trapping potential in a linear
Paul trap. The radial confinement (x-direction) results from fast oscillating voltages. The axial
confinement (z-direction) stems from a combination of the oscillating potential and static
fields. A harmonic confinement in all directions is achieved. (b) The funnel-shaped trap
geometry results in a tapered trapping potential. The equipotential planes provide an "egg"
shape. Such a trapping potential leads to a radial trap frequency which depends on the axial
position, and hence to a coupling between the radial and axial degree of freedom.
This oscillating potential has a tapered shape and depends on the angle θ. Urf and Udc
are the oscillating and static voltages on the trap electrodes, respectively, while r0 and
z0 are the distances of the electrodes from the center of the trap. The funnel-shaped
geometry of the trap design leads to a ponderomotive potential of the form
m (ω
2 2
0x x +ω
2
0y y
2)r40 m
Φ (x,y,z) = + ω2 z2p , (2.16)2 (r0 + z tan θ)4 2
0z
which is compared with the potential of linear trap in Fig. 2.4. As compared to a linear
trap, the modified trap design leads to a conical structure of the equipotential planes and
creates an effective coupling between the axial and radial degree of freedom. Comparing
Eq. (2.16) to Eq. (2.12), it results in radial trap frequencies ωr(z) that depend on the axial
position z according to
[ ]
tan θ −2
ωr(z) = ω0 1+ (z− z0) . (2.17)r0
In the heat engine experiments, this dependance is essential to implement compression
and expansion phases in the radial direction, while the ion moves back and forth along
the trap axis.
2.4.3. Origin and reduction of micromotion
Superimposed to the harmonic secular motion, an ion confined in a Paul trap may exhibit
a so-called micromotion [Lei03]. This is a fast oscillation which originates from the
time-dependent oscillating saddle potential in Eq. (2.10). Consequently, the micromotion
shows a frequency of Ωrf, which is generally much faster than the secular motion. The
21
2. General concept of the single-atom heat engine
amplitude of this micromotion depends on the distance r of the ion from the trap axis
and vanishes for r = 0. However, due to inaccuracies in the trap geometry or external
electric stray fields, the node of the effective trapping potential can deviate from the node
of the oscillating electric quadrupole field, resulting in a non-vanishing amplitude of
the micromotion. This excess motion can be compensated by applying additional static
electric fields in the radial directions.
Linear Paul-traps are usually operated in a configuration, where the electric rf-signal is
fed to two opposing trap electrodes, while the two remaining electrodes are connected to
ground or biased with a constant voltage, see Fig. 2.2. If the ion is not perfectly centered
with respect to the field lines, this leads to an additional micromotion in axial direction.
The reference voltage of the ion at the node of the trapping potential is oscillating with
the frequency of the rf-signal and half of its amplitude. As a consequence, the ion
experiences an effective rf-signal originating from the static voltage of the end-caps,
leading to an additional confinement in axial direction, as well as to micromotion in
this degree of freedom. Usually, for linear Paul-traps, this axial micromotion vanishes if
the ion is kept in the node of the oscillating saddle potential [Ber98]. However, in our
case, due to the tapered geometry, there is a strong component of the oscillating field in
axial direction, which leads to a macroscopic micromotion on the order of 10 µm for low
trap frequencies, see [Tol14]. Such a strong micromotion could conceal the expected heat
engine dynamics. In order to mitigate the axial micromotion, we apply an rf-signal to all
of the four electrodes: each pair of opposing electrodes is supplied with the same voltage
signal, V(t) = ±V0 sin(Ωrf t), while the sign is reversed between both pairs as sketched
in Fig. 2.3. This results in a static reference voltage of the ion at the node of the trapping
potential, which is equal to the reference level of the static and oscillating voltages. As
a consequence, the axial micromotion vanishes completely if the ion is centered at the
node of the oscillating quadrupole field.
2.5. Working cycle of the single-atom heat engine
The fundamental principle can be described by analyzing the forces acting on the ion.
In our case, the thermal energy during the operation of the engine is high compared
to the energy spacing of the quantized harmonic oscillator states (kBT ≫ h̄ωr, weak
binding regime), thus a classical description of the dynamics is appropriate. We assume
the ion to have minimized its potential energy in the trap, and thus all external forces
are in equilibrium. Heating and cooling processes lead to thermal states with different
temperatures T, providing a time-averaged spatial distribution described by
[
1 − ](r− r )2
ξr(r,T) =
0
2πσ2
exp . (2.18)
r (T) 2σ2r (T)
The time-averaged width σ(T) of this two-dimensional Gaussian probability distribution
depends on the temperature T according to
√
k T
σr(T) =
B
2 . (2.19)mωr
The heat engine is driven by alternately heating and cooling the ion in radial direction.
We assume to have heat baths at hand to change the temperature of the radial thermal
22
2.5. Working cycle of the single-atom heat engine
4
F
2 z x
0
-2
d
-4
-2 -1 0 1 2
Radial x,y (mm)
Figure 2.5. – Conversion of heat into work. (a) The ponderomotive potential in between the tapered
trap electrodes (gray). The thermal probability distribution of the ion (green, not to scale)
provides a finite and temperature dependent extent in radial direction. The radial restoring
force (orange arrows), which originates from the ponderomotive potential, has an axial
component for a non-zero radial distance from the trap axis, see text. (b) When a cold ion
(blue) is heated, the radial extent of the thermal distribution increases and therefore the axial
component of the radial restoring force rises. This leads to a shifted equilibrium position for a
hot ion (red) as compared to a cold ion (blue).
state of the ion in a controlled way. Later, in the experimental part in Sec. 6.1, we will
prove that this is achieved by exposing the ion to a combination of laser cooling and
electric field noise [Tur00a]. Heating and cooling changes the spatial width σr(T) of
the radial thermal state ξr(r,T). Owing to the geometry of the trap, the gradient of
the funnel-shaped pseudo-potential Φp has a component in the axial direction z, that
depends on the radial position r, see Fig. 2.5 (a).
Due to the tapered ponderomotive potential the restoring force F for an ion at a non-
zero radial distance r from the trap axis has a component in axial direction, which rises
with increasing r. It is obtained as the derivative of the ponderomotive potential Φp,
−dΦp(r,z)Fz(r > 0) = 6= 0. (2.20)dz
As a consequence of the width of the radial thermal state ξr(r,T), an ion centered at
the trap axis is exposed to an additional force in axial direction. Consequently, the
equilibrium position of the ion is shifted along the trap axis, see Fig. 2.5 (b). By integrating
over the radial spatial distribution we obtain the temperature dependent force
∫ 2π ∫ ∞ dΦp(r,z) 4 k r4 tan θ
F Bz(T) = − ξr(r,T) r drdφ = 0 T. (2.21)
0 0 dz (r0 + z tan θ)5
This axial force Fz(T) is the basis of the conversion of a temperature difference into
mechanical motion.
In the following, we analyze the dynamics of the ion when driven in a four stroke
thermodynamic cycle, as shown in Fig. 2.6. During the first part of the cycle the ion
is heated and the width σr(T) increases. As a result, during the second step, the ion
moves along the z-axis to a weaker radial confinement. The displacement within the axial
potential increases the total potential energy of the ion and work is produced. During
23
Axial z (mm)
2. General concept of the single-atom heat engine
Figure 2.6. –Working principle of the single-ion heat engine driven in four consecutive strokes of
an Otto cycle. (A) An initially cold ion is heated and the width of its probability distribution
expands. (B) The increased spatial width in radial direction leads to a shift of the ion along
the trap axis to the open end of the trap. (C) Cooling the ion reduces the radial width of its
probability distribution. (D) Due to the restoring force of the axial potential, the ion moves
back to its initial position. The thermodynamic cycle is closed and may be repeated.
the third step the ion is cooled to the initial temperature. The radial width σr(T) and the
corresponding force Fz decreases. Owing to the restoring force of the axial potential the
ion moves back to its initial position.
The combination of the heating and cooling processes give rise to a closed thermo-
dynamic cycle, see Fig. 2.6. However, a typical heating and cooling of ∆T = 10−2K
leads to a force in axial direction of the order of Fz = 10−22N, with a corresponding
static displacement of the order of z = F 2z/(mωz) = 10
−8m. However, it is desirable
to increase the displacement z of the ion, in order to achieve stronger compression and
expansion of the radial mode, as well as to facilitate the detection of the axial motion. To
this end, we switch between heating and cooling periods with precise timing, such that
the cycle repeats itself at a rate equal to the axial trap frequency ωz. Due to Eq. (2.21),
the resulting modulation of the temperature T(t) leads to a periodic force Fz(t) in the
axial direction. The engine therefore effectively drives a harmonic oscillation in the axial
direction. The work produced in each cycle is thus resonantly transferred to the axial
degree of freedom and the energy is stored in the amplitude of the z-oscillation.
However, the essentially frictionless nature of the system leads to an ever increasing
oscillation. In order to contain this oscillation, we provide adjustable damping to the
axial mode. A steady state operation is reached when the work generated by the engine
is balanced by the energy dissipated by the controlled damping.
For future applications and more complex engine systems, the storage of the generated
work in an independent degree of freedom is very important. This storage mode can
be seen as a flywheel, as sketched in Fig. 2.7, where other oscillatory systems may be
coupled to, in order to be driven by the engine [Aba12].
2.6. Laser cooling
One of the most fundamental processes within this work is laser cooling and the inter-
action of a trapped ion with coherent radiation. Photons with an energy Eγ = h̄ω can
drive transition of the electrons between individual energy eigenstates of the atomic
shell, if Eγ matches the transition energy ∆E. If the ion is in motion with a velocity
v, the Doppler effect shifts the resonance frequency ω0 by ∆ω = ω0 v/c, with c being
24
2.6. Laser cooling
Figure 2.7. – Sketch of the mechanical analog of the single atom heat engine: a cylinder with a piston
and a flywheel. The working agent is the radial mode of the ion (corresponding to the gas
mixture), interacting with engineered reservoirs (ignition and thermalization). The expansion
of the radial mode leads to a displacement of the ion in axial direction (piston). The work
produced is stored in the energy of the coherent axial oscillation (flywheel).
the speed of light. By detuning Eγ, only the ions with the appropriate velocity class
are resonant to Eγ. We can tune the frequency of the laser such that photons are only
absorbed, if the ions are moving towards the laser source. Thereby, each absorption
process reduces the momentum p of the ion by the photon momentum, p′ = p − h̄k.
The subsequent spontaneous emission is isotropic, and thus the momentum transfer
averages out after many scattering events. The result is a reduced kinetic energy of the
ion. Since this cooling process is based on Doppler-shifted resonances, it is often referred
to as Doppler cooling. For more details of the fundamental processes of Doppler cooling,
refer to [Ste86a] and [Met99].
As the ion is trapped in a harmonic potential, it performs oscillations and thus a
single laser beam is sufficient to cool all degrees of freedom [Lei03]. To cool an ion from
room temperature to rest, on the order of 104 photon scattering events are needed. This
number reveals the need for short lived cycling transitions for laser cooling thermal
ions efficiently. However, a short lifetime τ causes a broad resonance of the electronic
transition, Γ = 1/(2πτ). This limits the effective cooling for ions with low temperatures,
since ions moving in the same direction as the photons provide a finite absorption
probability and therefore are accelerated. For a two level system, and neglecting the
linewidth of the lasers as well as micromotion, the highest cooling rate is obtained for a
detuning of the laser frequency from resonance of ∆ω/2π = −Γ/2, and the resulting
minimum temperature, the Doppler cooling limit, is [Ste86a]
h Γ
TD = , (2.22)2 kB
where kB is the Boltzmann constant. However, in most experiments, the atomic level
structure is more complex than a two level atom, the laser linewidth is not negligible and
micromotion generates sidebands of the resonances. As a consequence, the theoretical
limit TD is usually not reached [Cir94a].
In order to cool trapped ions further down towards the ground state of motion,
resolved sideband cooling or electromagnetically induced transparency cooling may be
employed [Die89, Roo00, Lei03].
25
2. General concept of the single-atom heat engine
Figure 2.8. – The relevant part of the energetic level scheme of 40Ca+ reduces to a three level system.
The individual levels are shown together with their radiation lifetime and their total energy
in terms of their wavenumber. The transitions between the indicated levels are shown as
well. The colored transitions are driven by light fields. The spacing of the energy levels is not
drawn to scale.
2.7. The calcium ion
In order to reduce the experimental complexity and to achieve good conditions for
laser cooling, earth-alkali ions are usually used in Paul-trap experiments. Their level
scheme is similar to alkali-atoms, having one single valence electron in an S-state above
a closed shell. Additionally, atoms without nuclear spin and thus without hyper-fine
structure are favorable. In our experiments we use 40Ca+ ions, which have an electronic
ground-state configuration of the closed electron shell of Argon with one additional S
electron, [Ar] 4s 2S1/2. The relevant part of the energetic level structure can be reduced to
a three-level system with fine structure levels 4s 2S1/2, 3d 2D3/2 and 4p 2P1/2, see Fig. 2.8.
The short lived S1/2 ↔ P1/2 dipole transition with a lifetime of τP = 6.9 ns [Het15] is
used for laser cooling and is driven by a light field near 397 nm. The short lifetime τP
results in a natural linewidth of Γ −1SP = [2πτP] = 22.1MHz. However, the ion decays
from the P1/2 state with a probability of p = 0.064 to the metastable state D3/2 [Ram13].
The S1/2 ↔ D3/2 transition is a quadrupole transition, and thus provides a long lifetime
of 1.18 s [Kre05]. In order to reach a closed cycling transition for laser cooling all the
population which decays into D3/2 is pumped back to the P1/2 state by a light field
near 866 nm. If a magnetic field is applied, the individual magnetic sub-levels mj split ac-
cording to the Zeeman effect. Individual transitions between the mj levels are addressed
separately by choosing the appropriate polarization of the laser beams with respect to
the quantization axis of the magnetic field.
26
3 Results of analytical and numerical
investigations
In this chapter we present analytical and numerical investigations of the single-ion heat
engine dynamics. Due to its conceptual simplicity, we consider a quantum Otto cycle
where the bath interactions and the expansion and compression phases are strictly sepa-
rated. Therefore, we simulate a single trapped ion confined in a tapered potential and
coupling it alternately to two engineered thermal reservoirs. We use realistic experimen-
tal parameters, to be as close to a real experimental environment as possible. As for
all realistic machines, the thermodynamic Otto cycle runs in finite time and therefore
the engine has non-zero output power [And84]. We will show that our device converts
incoherent thermal motion into coherent regular motion, thus fulfilling the fundamental
definition of heat engines.
In the first part, starting with Sec. 3.1, we will present analytical derivations of the
key parameters of the Otto cycle, based on the framework of the quantum harmonic
oscillator. The latter plays an important role in theoretical investigations of thermody-
namic quantum devices, e.g. [Lin03, Rez06, Qua07]. We will show how to determine the
output power and the efficiency at maximum power in the limit of adiabatic processes.
A detailed analysis for non-adiabatic processes can be found in [Aba15]. The analytical
derivations are fully valid for the quantum regime, where the discreteness of the energy
spectrum plays an important role. However, it can applied to classical mechanics in the
hot temperature limit without loss of generality, following the Ehrenfest theorem [Ehr27].
We further present semi-classical Monte Carlo simulations in Sec. 3.2, that demonstrate
the experimental feasibility of the single-ion heat engine and allows us to identify experi-
mental constraints. In general, all of the parameters of our engine system are tunable over
a wide range, in particular the temperatures of the heat baths and the trap frequencies.
In this idealized system the maximum efficiency can be achieved over a large span of
temperatures. This first part is based on our publication
Single-ion heat engine at maximum power,
O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer and E. Lutz,
Phys. Rev. Lett. 109, 203006 (2012).
In the second part of this chapter, starting with Sec. 3.3, we will extent our findings
to non-thermal heat baths. Again, we will present both, analytical investigations and
Monte-Carlo simulations of the system. We will demonstrate by the example of squeezed
thermal reservoirs that the efficiency at maximum power can be increased beyond the
27
3. Results of analytical and numerical investigations
standard Carnot-limit, which is based on purely thermal reservoirs. These findings
manifest the potential of non-thermal heat baths in the context of nano heat engines
and show how different energy sources can be used simultaneously to drive nanoscopic
engines and to increase their output power. In order to consider the constraints of the
second law of thermodynamics we will deduce a generalized Carnot bound, which limits
engine systems with non-thermal reservoirs and takes the increased total energy of the
reservoirs into account. This second part is completed by an extended paragraph about
the interpretation of our findings on heat engines with non-thermal reservoirs, Sec. 3.3.4.
The results in this part are based on our publication
Nanoscale Heat Engine Beyond the Carnot Limit,
J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer and E. Lutz,
Phys. Rev. Lett. 112, 030602 (2014).
3.1. The quantum harmonic oscillator as heat engine
We start with an analytical formalism to fully describe the key parameters of an Otto
engine based on a quantum harmonic oscillator. This formalism starts from the mean
energies of the harmonic oscillator after each of the four strokes. These energy states fully
characterize the dynamics of the engine and allow to calculate all relevant parameters, in
particular the efficiency at maximum output power.
3.1.1. Thermal state of a quantum harmonic oscillator
We assume a one dimensional harmonic oscillator with the mean number of phonons n.
The classical and quantum representations of the Hamiltonian of this system are
2 ( )1 2 mω 1Ĥc = p̂ + x̂2 and Ĥ = h̄ω ↠â+ , (3.1)2m 2 q 2
respectively, with n = 〈n| ↠â |n〉. The Schrödinger equation Ĥ |Ψ〉 = E |Ψ〉 reveals the
energy of the phonon states
En = h̄ω(n+ 1/2). (3.2)
The harmonic oscillator is in a thermal state if the phonon state |n〉 is occupied with a
probability proportional to the Boltzmann factor
1
P = e−βEnn (3.3)Z
with the inverse temperature β = 1/(kBT) and a normalization factor Z [Sch01, Ger05].
When measuring n of the quantum system in thermal equilibrium many times, the mean
phonon number results to [Mee96, Lei03]
〈n(β,ω)〉 1= , (3.4)
exp [βh̄ω]− 1
also known as the Planck distribution [Sch01]. The temperature T of the system is then
defined as
T(〈n〉,ω) = (h̄ω
k ln 〈
) . (3.5)
n〉+1
B 〈n〉
28
3.1. The quantum harmonic oscillator as heat engine
W
n 2 D C
3
Q 4 2 Q
1
n 1 A B
W
ω1 ω2
Frequency
Figure 3.1. – Otto cycle for a harmonic oscillator in the {n,ω}-representation. During the isentropic
strokes 1 and 3, the frequency of the oscillator is varied between ω1 and ω2 at constant phonon
number distribution. During the isochoric strokes 2 and 4, the temperature of the oscillator is
changed at constant frequency between β1 and β2 and the mean phonon number is changed
between 〈n〉1 and 〈n〉2. The individual states of the ion in the harmonic potential are sketched.
While the engine consumes work during stroke 1, it produces work during stroke 3. Due to
the higher total energy of stroke 3 the net work of the cycle is positive and equals its area.
Using the mean phonon number Eq. (3.4) and the energy of the harmonic oscillator
Eq. (3.2), we obtain the mean energy for a thermal state
( )
〈 h̄ω βh̄ωH〉 = coth , (3.6)
2 2
where we have used the trigonometric replacement coth(x) = 1+ 2 [exp(2x)− 1]−1.
3.1.2. Otto cycle for a harmonic oscillator
In order to drive the quantum harmonic oscillator in a thermodynamic cycle, the pa-
rameters β and ω become functions of time. Compared to macroscopic engines, the
trap frequency corresponds to the volume of the system, while the thermal energy is
related to its pressure. The engine is now coupled alternately to two heat baths at inverse
temperatures βi = 1/(kBTi) with (i = 1,2). The Otto cycle for the harmonic oscillator
consists of four consecutive steps, which are analogue to the strokes of a classical Otto
engine:
1. Isentropic compression: A(ω1,β1) → B(ω2,β1): the frequency is varied during time
τ1 while the system is isolated. The evolution is unitary and quasistatic. The
phonon number distribution and the von Neumann entropy of the oscillator are
constant.
2. Hot isochore: B(ω2,β1) → C(ω2,β2): the oscillator is weakly coupled to a reservoir
at inverse temperature β2 at fixed frequency and allowed to relax during time τ2 to
the thermal state C.
3. Isentropic expansion: C(ω2,β2) → D(ω1,β2): the frequency is changed back to its
29
Mean phonon number
3. Results of analytical and numerical investigations
initial value during time τ3. The isolated oscillator evolves unitarily and quasistati-
cally at constant entropy and constant phonon numbers into state D.
4. Cold isochore: D(ω1,β2) → A(ω1,β1): the system is weakly coupled to a reservoir
at inverse temperature β1 > β2 and relaxes to the initial thermal state A during τ4.
The frequency is again kept constant.
The mean energies 〈H〉 of the oscillator at the four stages of the cycle are [Aba15]
( )
〈H〉 h̄ω1 β h̄ωA = coth 1 1 , (3.7a)2 ( 2 )
〈 h̄ω β h̄ωH〉B =
2 coth 1 1 , (3.7b)
2 ( 2 )
〈 〉 h̄ω2 β h̄ωH = coth 2 2C , (3.7c)2 ( 2 )
〈 〉 h̄ωH = 1 β2h̄ωcoth 2D , (3.7d)2 2
This set of equations is valid for systems in the quantum regime, βh̄ω ≫ 1, as well
as in the classical limit, βh̄ω ≪ 1. In the latter the expressions simplify according to
coth(x) ≈ 1/x.
The single-ion system represents a harmonic oscillator which is in general thermally
isolated from its environment. In the limit of adiabatic processes, the probability dis-
tribution for phonon states, Eq. (3.3), remains unchanged, while the frequency ω and
thus the energy level spacing is changed [Sch01]. Note that adiabatic in the quantum
thermodynamic context means that the individual transitions occur isolated from their
environment and quasistatically. Consequently, there is no heat exchange or thermaliza-
tion and the transitions are much slower than the timescales of the internal dynamics,
which are on the order of 1/ω. As we will show later, in the experimental realization
the timescale of the thermodynamic transitions is set by the axial trap frequency 1/ωz,
while the timescale of the internal thermalization processes is on the order of the radial
trap frequency 1/ωr. The assumption of quasistatic processes holds as long as ωz ≪ ωr.
After thermalization with the heat baths, the system is in thermal equilibrium at stages
A and C.
However, as a result of the adiabatic and isentropic processes, at stages B and D the
system is still distributed thermally with respect to A and C, respectively, while the level
spacing has changed. As a consequence, the states B and D are non-thermal states and,
in contrast to macroscopic systems, the engine operates far from equilibrium [Def13].
Now, we want to find a well suited and meaningful representation of the Otto cycle. In
the case of an isolated harmonic oscillator, the relevant pair of conjugate variables is the
frequency ω (intensive) and the mean phonon number 〈n〉 (extensive), whose product
equals its internal energy [Lin03]. The Otto cycle of the harmonic oscillator is sketched
in Fig. 3.1 in the {n,ω}-representation. The total internal energy changes during the each
of the transitions, either by changing the probability distribution Eq. (3.3) and thus 〈n〉,
or by changing the frequency ω; always keeping the other variable constant,
dE = h̄〈n〉 · dω+ h̄ω · d〈n〉. (3.8)
30
3.1. The quantum harmonic oscillator as heat engine
As shown in [Lin03], the change of the internal energy can be compared directly to the
differential form of the second law of thermodynamics
dE = dW + dQ. (3.9)
By comparing Eq. (3.8) and Eq. (3.9), we identify the work produced dW and the heat
exchanged dQ as
dW = h̄〈n〉 · dω and dQ = h̄ω · d〈n〉. (3.10)
As a consequence, Eq. (3.8) is a representation of the second law of thermodynamics
for a harmonic oscillator, and the area of the {n,ω}-diagram reveals directly the work
produced during each cycle.
However, often it is more convenient for practical reasons to choose a different repre-
sentation basis for the thermodynamic cycles. It is widespread to use an energy-frequency
representation {H,ω}, since 〈H〉 is actively changed during the strokes and contains all
information about the thermal and the mechanical properties. The same applies here. In
the following we will show how to calculate all relevant thermodynamic quantities, such
as heat, work and efficiency, from the total energy 〈H〉 of the four states of the system,
Eqs. (3.7a-3.7d).
3.1.3. Work and efficiency
In order to determine the efficiency of the quantum Otto cycle, we need to evaluate work
and heat for each of the above steps. During stroke 2 and 4, the frequency is constant,
and thus only heat is exchanged with the reservoirs. On the other hand, during stroke 1
and 3, the system is isolated and only work is performed by modulating the frequency.
Let us now derive the mean values for heat and work during the individual steps. The
mean work during the first stroke is
( ) ( )
〈 〉 〈 〉 − 〈 〉 h̄ω2 − h̄ω1 β h̄ωW1 = H B H A = coth 1 1 . (3.11)2 2 2
The mean heat exchanged with the hot reservoir during the second stroke reads
[ ( ) ( )]
〈Q2〉 〈 〉 − 〈 〉
h̄ω2 β2h̄ω2 β1h̄ω= H C H
1
B = coth − coth . (3.12)2 2 2
In a similar way, the average work and heat for the third and fourth stroke are given by
( ) ( )
〈W3〉 = 〈H〉D − 〈 〉
h̄ω1 − h̄ω β h̄ωH C =
2 coth 2 2 (3.13)
2 2 2
and
[ ( ) ( )]
〈 〉 〈 〉 − 〈 〉 h̄ω1 β h̄ω β h̄ωQ = H H 1 14 A D = coth − coth
2 2 . (3.14)
2 2 2
For a heat engine operation, heat is absorbed from the hot reservoir 〈Q2〉 ≥ 0 and
flows into the cold reservoir 〈Q4〉 ≤ 0. Regarding the mechanical energy of the en-
gine, work is consumed by the system during the first stroke 〈W1〉 ≥ 0, while it pro-
duces work during the third stroke 〈W3〉 ≤ 0. The net work of the system is negative,
31
3. Results of analytical and numerical investigations
a) 0.20 b)
β2 /β1 = 0.5 Wmax
β2 /β1 = 0.6
0.15 β2 /β1 = 0.7
β2 /β1 = 0.8
η* 0.4
0.10
0.0
0.2
0.05
0.5
0.0
1.0
0.00
0.0 0.1 0.2 0.3 0.4 0.5 0.5
Effic 1.0Efficiency η iency η 0.0
c) 1.0 d) 2.0
η*
ηC
0.8
1.5
0.6
1.0
0.4
0.5
0.2
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Temperature ratio β2 /β1 Temperature ratio β2 /β1
Figure 3.2. – Analysis of work and efficiency of the Otto cycle in the high temperature limit for
different ratios of the bath temperatures β2/β1. (a) Mean work 〈W〉ht(η) in terms of kB
as a function of the efficiency η for different constant values β2/β1. For η∗ the generated
work reaches its maximum, 〈W〉max. (b) 3D representation of the mean work 〈W〉ht(η) for
continuous values of β2/β1. The curves in a) correspond to vertical intersections along
the colored lines. Efficiencies above the Carnot limit ηC = 1− β2/β1 (red shaded area) are
excluded by the second law of thermodynamics. (c) Efficiency η∗ as a function of β2/β1, which
corresponds to the ridge of the work surface, red line in b). The curves from a) are indicated
by dashed lines. The Carnot efficiency for thermal machines ηC serves as comparison. (d)
Maximum mean work 〈W〉max corresponding to η∗, normalized to 1/β2, as a function of
β2/β1, cf. red line in b). For β2/β1 → 0 the temperature difference diverges and so 〈W〉max
does.
〈W〉 = 〈W1〉+ 〈W3〉 ≤ 0, and thus the system generates more work than it consumes, as
expected for heat engines.
The efficiency of the harmonic oscillator engine, defined as the ratio of the total work
per cycle and the heat received from the hot reservoir, then follows as
−〈W1〉+ 〈Wη = 3〉 ω〈 〉 = 1−
1 . (3.15)
Q2 ω2
For practical purposes, we search for the maximum output power for given bath temper-
atures, and for the corresponding efficiency at maximum power. In the high temperature
limit, where βh̄ω → 0 and coth(x) ≈ 1/x, the net work reads
( ) ( )
〈 〉 〈 〉 〈 〉 1 ωW = W + W = 2 1 ωht 1 3 − 1 + 1 − 1β1 (ω1 β)2 ω2
1
= η − β1 η . (3.16)
β1 β2 1− η
In Fig. 3.2 (a) we plot 〈W〉ht as a function of the efficiency η, for different constant tem-
32
Efficiency η Work W (kB)
Work Wmax  (1/β2)
 (kB)
Work W
Temp. ratio β
2 /β
1
3.2. Proof of concept - a single ion as working fluid
perature ratios β2/β1. Analytically, we find the maximum of these curves by optimizing
〈W〉ht with respect to ∆ω = ω2 −ω1, while keeping all parameters but ω2 constant,
√
d〈W〉ht = 0 ⇒ ω2 β= 1 . (3.17)
dω2 ω1 β2
Applying this condition to the expression for the efficiency Eq. (3.15), we find the effi-
ciency at maximum power output
√
η∗
β
= 1− 2 , (3.18)
β1
which corresponds to the well known Curzon-Ahlborn efficiency for classical systems,
Eq. (2.9), see also [Aba12]. We have thus shown that the quantum energy representation
of the four states of the heat engine agrees well with the classical expectations in the high
temperature limit. Fig. 3.2 illustrates the behavior of η∗ and of the corresponding work
output 〈W〉max as a function of β2/β1.
Eq. (3.18) indicates that maximum efficiency of 1 may be attained when β1 → ∞. In
this low-temperature limit for the cold bath, quantum effects become crucial. Repeating
the above optimization in the regime β1h̄ω1 → ∞ with coth(∞) = 1, we find for the
efficiency at maximum power
√
h̄ω β
η∗ = 1− 1 2q (3.19)2
√
for ω2 = 2ω1/h̄β2. Here, the classical thermal energy kBT1 is replaced by the ground
state energy of the oscillator h̄ω1/2. The above expression is the quantum extension of
the Curzon-Ahlborn efficiency.
Note, that a single particle engine exhibits statistical fluctuations in all of the thermo-
dynamic variables from cycle to cycle, in contrast to thermodynamic systems based on
large ensembles of particles. Therefore, in the following, we employ the mean values
of the thermodynamic variables. The features of fluctuations are extensively discussed
in [Din16].
3.2. Proof of concept - a single ion as working fluid
In this section, we demonstrate the feasibility of the implementation of the thermody-
namic cycle of a heat engine for a single trapped ion, as introduced in Sec. 2.5. Therefore,
we present numerical results of semiclassical Monte-Carlo simulations of the trapped
ion system driven in an Otto cycle. We employ realistic experimental parameters and the
tapered trap geometry, which has been presented in Sec. 2.4.2. Later in Sec. 6.1.2, we will
prove experimentally that the interaction with thermal reservoirs can be mimicked by ei-
ther laser cooling (cold reservoir) or a combination of electric field noise and laser cooling
(hot reservoir) [Tur00b]. Both interactions are modeled by Monte-Carlo simulations.
Since we aim to achieve realistic experimental conditions, we model the bath coupling
by simulating the interaction with laser photons and electric field noise in a realistic
fashion, instead of changing the temperature of the ion by an interaction with a virtual
perfect heat bath or by a direct adjustment of the phase-space variables. In this way, we
will demonstrate the functionality of our experimental approach. Furthermore, we will
investigate how the maximum work 〈W〉max of the engine may be achieved.
33
3. Results of analytical and numerical investigations
3.2.1. Concepts of numerical simulations
In order to solve the equations of motion of the trapped ion under realistic conditions, we
first have to model the geometry of the trapping potential inside the trap, as explained
in detail in [Sin10]. To obtain the time variable electric fields inside the trap volume
the Laplace equation ∆Φ(~x) = 0 has to be solved for each point in space with Dirich-
let boundary conditions Φ(~x) = Ui on the surface of the ith electrode. To reduce the
computational requirements we employ the Boundary Element Method (BEM) [Poz02]: we
discretize the surface of the electrodes into small rectangular-shaped elements and calcu-
late the electric potential at any point in space as a superposition of the contributions of
all elements with surface charge Qi, using Coulomb’s law Φ(~x) = ∑i Qi/(4πǫ0|~x− ~xi|).
To simulate the dynamics of the ion, we model radio-frequency oscillations of the
electrode voltages and thus a time dependent potential Φ(~x,t). For a fast numerical
propagation of the ion trajectory we approximate the numerical solution of the potential
Φ(~x,t) by an analytical and time dependent description Φ′(~x,t), similar to Eq. (2.15). This
leads to a modified force on the ion ~F(~x,t) = −(q/m)∇~ Φ′(~x,t) after each time step h.
This equation of motion allows us to propagate the ion trajectory in phase-space. To
solve the equation of motion we use the Störmer-Verlet method, which is based on a
partitioned Runge-Kutta propagator [Gre06]. It is time invariant, energy and phase space
conservative and popular in molecular dynamics simulations [Ver67]. Thereby, position
~xn and velocity ~vn are propagated jointly, starting from ~x(t0) = ~x0 and d~x(t0)/dt = ~v0,
following the recursion [Sin10]
h
~vn+1/2 = ~vn + · ~Fn(~xn) (3.20)2
~xn+1 = ~xn + h ·~vn+1/2 (3.21)
h
~vn+1 = ~vn+1/2 + · ~F2 n+1(~xn+1), (3.22)
where the notion xn is a short form of x(tn), with tn = n · h.
The interaction of the ion with the laser beams is modeled by realistic Monte-Carlo
simulations. Therefore we modify the velocity ~vn+1/2 after Eq. (3.21) by the total mo-
mentum transfer of the photon absorption and emission process. First, a photon at the
frequency of the laser νL is absorbed with momentum p = h̄~L kL. During the subsequent
spontaneous decay, a photon is emitted at the frequency of the atomic resonance νL with
momentum psp = h̄~ksp Thus, ~v is modified by
(
′ h̄|kL| h̄|ksp|
)
~v n+1/2 = ~vn+1/2 + Psp · ~uL − ~usp . (3.23)m m
Here, ~uL is the direction vector of the laser beam and ~usp is a random direction vector
accounting for the isotropic spontaneous emission. The scattering probability Psp is
non-zero during periods where the lasers are on, and is calculated by [Cas88]
S
Psp = Γ · h · , (3.24)1+ 2 S
with the saturation parameter
Γ2
S = I · ( )2 , (3.25)
∆ν−~kL ·~v 2n+1/2 + Γ
34
3.2. Proof of concept - a single ion as working fluid
where Γ = 1/(2πτ) = 23.1MHz is the natural linewidth, I the relative laser intensity
with respect to the saturation limit and ∆ν the spectral detuning of the laser. In each
time step h a photon scattering event takes place if a random number is larger than Psp,
while h should be smaller than the lifetime τ of the excited atomic level (τ = 6.9 ns for
Ca+). Here, we have done several approximations: we reduce the level scheme to a two
level system, we ignore the linewidth of the laser and we neglect the time delay between
the absorption and emission event. For the heat engine dynamics with a high absolute
temperature of the ion, kBT ≫ h̄k, these approximations have a marginal effect and the
Monte-Carlo simulation of laser cooling leads to realistic results.
To heat the motional state of the ion we expose the ion to electric field noise. This
effect is simulated by adding a static electric field ~E in the radial (x,y)-plane to trapping
potential Φ′(~x,t). Amplitude and direction of ~E are randomly changed for each time
step, following a normally distributed probability for the amplitude with a variable
expectation value N .
According to the Monte-Carlo method, the probabilistic nature of a thermal state of
the ion is reproduced by a repeated sampling of several thousand classical trajectories of
the ion under randomized interactions [Cas88]: the initial parameters for each trajectory
are chosen randomly according to a Boltzmann distribution. The thermal probability
distribution during the experimental runs are realized through photon scattering and
electric noise interaction [Sre94]. The ensemble of all trajectories reveals a normally
distributed probability for position and velocity coordinates of the ion. In the following
this technique will be discussed in more detail.
3.2.2. Monte Carlo simulations
We reach a dynamic confinement of the ion in the trap volume through the oscillating
parabolic and tapered saddle potential Eq. (2.15) with a trap drive frequency Ωrf =
60MHz. We assume a funnel-shaped trap geometry, as sketched in Fig. 2.3, with an angle
of the rf-electrodes of θ = 20◦ with respect to the trap axis. The length of the trap is 5mm
and the radial distance of the rf-electrodes to the trap axis is r0 = 1mm at the position
of the ion. The axial and radial trap frequencies of the ion are ω0,z/(2π) ≃ 35kHz
and ω0,x/(2π) ≃ ω0,y/(2π) ≃ 6.0MHz. A small offset field in the x-direction lifts the
degeneracy of both radial oscillations. The time steps h of the numerical integration
are chosen much shorter than the oscillation period of the trapping field, h ≪ 1/Ωrf, in
order to resolve the micromotion of the ion.
The trapped ion is initially prepared in a thermal state at low temperature by ran-
domized initialization of position and velocity with Gaussian probability distribution
and consecutive laser cooling to the cooling limit in all spatial directions. The engine is
then driven by alternately heating and cooling the radial modes of the ion. As discussed
above, thereby we mimic the interaction with a hot and a cold reservoir. Both interactions
are oriented in radial direction, parallel to the (x,y)-plane. Therefore, only the radial
components of ~x and ~v are addressed by the thermal reservoirs. The axial degree of
freedom experiences a residual heating due to spontaneously emitted photons during
laser cooling of the radial modes. The coupling to the heat reservoirs is switched on and
off with a given period. A change in temperature T of the radial thermal state of the ion
leads to a modification in the axial component of the restoring force of the trap, which
changes the equilibrium position z0(T) (see Sec. 2.5). Heating and cooling the radial
state hence moves the ion back and forth along the trap axis, resulting in the closed Otto
35
3. Results of analytical and numerical investigations
Figure 3.3. – Simulated energy-frequency diagram of the Otto cycle driven in the radial mode of
the ion. The continuous line represents the ideal process, while the dots show the results of
the Monte Carlo simulations. Here we have plotted the mean of 105 individual runs of the
simulation. Remaining fluctuations due to the micromotion are eliminated by averaging each
point in time over one oscillation period of the trap drive 1/Ωrf. The curved shape of the
latter is due to the harmonic oscillation of the ion and the finite interaction time with the heat
baths. The pictograms illustrate the four individual strokes of the cycle.
cycle. A numerically simulated cycle is shown in Fig. 3.3. The energy gained by each
cycle in the radial degrees of freedom is stored in the axial motion. The energy transfer is
maximized, if the coupling of the ion to the thermal reservoirs is resonant with the axial
trap frequency. Therefore, the switching of the heating and cooling is adjusted to match
ωz. Each interaction is acting on the ion for 20% of an axial trap period. The ensemble
of driven oscillators is thus excited coherently in axial direction, such that heating and
cooling takes place at the turning points of the trajectory. In such a way, the cyclic
temperature change of the radial thermal state is directly converted into an increasing
coherent axial oscillation. By a variation of the noise amplitude N , radial temperatures
in the range of 20 to 200mK were achieved, corresponding to 0.1 < β2/β1 < 1 and
respective radial phonons numbers of about 400 and 4000. For a stable maximum axial
amplitude of about 1mm, the relative variation of the radial frequency at 6.0MHz is in
our setup about 50%.
To get a better understanding of the dynamics, we focus on the individual parameters
of the thermodynamic cycle in Fig. 3.3, the radial energy and the radial frequency in
terms of the axial position. Fig. 3.4 investigates the evolution of the total energy of the
radial degrees of freedom, E = m/2 (v2 + ω2 2r r r xr ). It shows how the system evolves
towards a steady state between heating and cooling. A detailed analysis of the energy
changes allows to identify heating and cooling processes, as well as the change of the
total energy due to the variation in the trap frequency. During the latter periods work is
performed by the engine. The motion of the ion in all spatial directions is analyzed in
Fig. 3.5. The undirected, statistical and thermal motion in radial direction is converted
into a regular and coherent motion of the ion in axial direction, which is the central
36
3.2. Proof of concept - a single ion as working fluid
Figure 3.4. – Simulated evolution of the radial energy during heat engine operation. (a) First, the
ensemble is randomly initialized at an average energy of 〈Hr〉 = 2× 10−24 J. Subsequent
laser cooling reduces the energy to a steady state between photon absorbtion and emission
processes. When starting the engine, first the cooling process is less effective than heating due
to the small velocities. This changes with increasing energy in favor of the cooling efficiency.
At 〈Hr〉 ≈ 3× 10−24 J, a steady state is set up between heating and cooling. (b) A detailed
analysis of the energy evolution in steady state allows to identify heating and cooling periods
as well as isentropic compression and expansion phases.
Figure 3.5. – Ensemble average of several thousand ion trajectories during heat engine operation.
The engine is started at t = 3ms; at t = 18ms the axial dissipation confines the amplitude
in a steady state. (a) Radial component. After the start of the engine an increase in the total
variance of the ensemble average is observed, while statistical fluctuations are dominant. (b)
On a scale of the oscillation period 1/ωz the statistical fluctuations are still dominant. (c)
Axial component. The ensemble average reveals a phase coherent superposition of regular
oscillations with increasing amplitude. The artificial dissipation limits the amplitudes to some
tens of micrometers. (d) A magnification of the trajectory reveals the regular oscillation. The
resolved trajectories (b) and (d) illustrate the conversion from incoherent into coherent motion.
37
3. Results of analytical and numerical investigations
Figure 3.6. – Phase space distribution of the motional state at different phases during an engine
cycle: the radial mode is changing from thermal to non-thermal states, always centered at the
origin of the coordinates. In axial direction, a coherent oscillation with typical displacement
of α = 103 ground state wave packet sizes displays the functional performance of the engine.
working principle of any heat engine.
An analysis of the ensemble average in phase space illustrates this working principle.
Due to the isentropic compression and expansion phases the phase space distribution of
the radial components changes between thermal and nonthermal states, but, as shown in
Fig. 3.6, it is always centered at the origin of the coordinates. Due to the mode coupling a
change in the variance of this centered distribution is converted into a displacement of
the distribution of the axial mode, which then performs a phase synchronized oscillation.
The increasing axial oscillation is only limited by the trap geometry. However, to reach a
steady state of the oscillation, a red-detuned low intensity dissipation laser is applied in
axial direction that damps the coherent movement and removes the excess heating from
spontaneous emission events.
In our setup, the oscillation amplitude in steady state as well as the trap frequencies
of the ion can be changed by virtue of the laser and trap parameters precisely. This
unique flexibility of the device can be exploited to satisfy the optimality condition for
maximum output power derived previously. Fig. 3.7 shows that by properly adjusting
the parameters our simulated Otto engine is able to run at maximum efficiency at
maximum power in an interval 0.5 < β2/β1 < 1. The maximum efficiency, determined
through Eq. (3.15) with values taken from simulated cycles as shown in Fig. 3.3, is about
η∗ ≈ 30%. This maximum efficiency is significantly larger than those obtained to date
experimentally [Ste11, Bli12]. The expected net power of the engine is of the order of P ≃
10−20J/s. Wewill show in the experimental part of this thesis, that in real experiments this
idealized efficiency is not reached yet due to significant technical limitations. However,
these limitation are of technical nature rather than physical restrictions.
3.3. Employing non-thermal reservoirs
Among the different kind of thermodynamic machines, we find two major classes. First
we have the well known combustion engines, which operate between two heat baths
at different temperatures T1 < T2 and convert the temperature difference into mechan-
38
3.3. Employing non-thermal reservoirs
1.0
η*
ηC
0.8
0.6
0.4
●
●
0.2
● ● ●
●
0
0 0.2 0.4 0.6 0.8 1.0
Temperature ratio β2 /β1
Figure 3.7. – Efficiency η of the Otto engine as a function of the temperature ratio β2/β1. The
black line the Carnot efficiency ηC with highest possible η but zero output power. The red
line corresponds to the theoretical adiabatic process with maximum output power and the
efficiency η∗, Eq. (3.18). The black points are the results of the numerical simulations and
demonstrate that the engine runs at maximum power. The green shaded area indicates the
accessible parameter range for realistic conditions, including the limited dimensions of the
trap, the laser beams and the detection devices, with an upper bound at ω2 ≈ 1.5ω1.
ical work [Cal85, Cen15]. Then there are molecular motors, which operate with one
single heat bath, the ambient temperature [Jül97, How01, Rei02a]. They are driven by
a variation of some external parameters. We find those kind of engines in biological
motor proteins as well as in artificial nano machines [Scu03, Kay07, Hän09, Leg15]. The
efficiency of thermally driven classical combustion heat engines is generally bound to the
Carnot efficiency ηC [Cen15]. The Carnot formalism is a direct consequence of the second
law of thermodynamics and is based on purely thermal and equilibrium reservoirs. How-
ever, when transferring this formalism to non-thermal or non-equilibrium systems, the
fundamental assumptions of Carnot do no longer hold and deviations are expected. For
molecular motors, an extreme form of non-thermal machines, the efficiency may in prin-
ciple reach unity η ≃ 1 [Par99, Ast07, Toy10, Toy11], and even the efficiency at maximum
power may reach η∗ ≃ 1 for strong driving [Sei11, Bro12]. Different theoretical studies
have already shown that heat engines may exceed the Carnot bound when coupled to en-
gineered non-equilibrium resources, such as quantum coherent [All00, Scu03], quantum
correlated [Dil09] or non-Markovian reservoirs [Zha14b]. Other examples of quantum
heat engines are masers or photocells, which are classically limited by the Carnot effi-
ciency as a consequence of the detailed balance between absorption and emission. Here
as well, different approaches show that this classical limit can be overcome by exploiting
their quantum properties [Mil03, Scu10, Nal13, Dor13]. All of these systems are charac-
terized by a temperature as well as additional parameters that quantify the degree of
deviation from pure thermal states. However, they all are consistent with the physics
behind the second law of thermodynamics, since the total entropy change of the system is
still positive, ∆S ≥ 0, as shown in [Bou13]. In order to respect the additional information
included in non-classical reservoirs, the second law of thermodynamics has to be revised
for those system, leading to a generalized Carnot bound [Aba14, Woo15, Gar15]. For
micro- and nanoscopic engines the thermodynamic observables are no longer deter-
39
Efficiency η
3. Results of analytical and numerical investigations
mined by their mean values, but become highly fluctuating quantities, even in thermal
equilibrium. Thermal fluctuations of energy fluxes can lead to statistical violations of the
Carnot bound on short timescales, while the second law of thermodynamics holds for
the long time average of the system. Those statistical short term violations of the second
law have been predicted theoretically [Kie04, Ver14a, Ver14b, Cam15] and demonstrated
experimentally [Wan02, Mar16].
In this section, we extend the general theory of a thermal (quantum) heat engine from
Sec. 3.1.2 to the case where it is coupled to a non-thermal reservoir. The fundamental
motivation of our studies is to demonstrate that non-thermal reservoirs provide certain
features that are not accessible with classical engine systems. Our investigations were
motivated by the recent advances in experimental reservoir engineering techniques,
such as in ion traps [Poy96, Mya00, Lin13, Kie15], in optical lattices [Die08, Ver09], in
circuit QED [Mur12, Sha13b] and in microwave cavities [Pie07, Sar11]. We evaluate
both the efficiency and the efficiency at maximum power of the single-ion engine. We
show that the efficiency at maximum power can be increased beyond ηC by exploiting
the non-thermal properties of the reservoir, without questioning the universality of the
general framework of thermodynamics. However, by exceeding ηC we demonstrate that
non-thermal reservoirs can achieve efficiencies, even for maximum work output, which
are not accessible for engines operating in thermal equilibrium.
Explicitly, we investigate the influence of non-thermal reservoirs on heat engines on the
example of a squeezed thermal reservoir. The reason for this choice is the experimental
realizability and the simplicity in the theoretical description of its dynamics. Squeezing
is a general concept in quantum optics [Scu97]. It may be characterized by a parameter κ
such that the phase-space quadratures of a state are, respectively, multiplied by eκ and
e−κ [Tei89], see Fig. 1.3 (d). Squeezed ground states of the harmonic oscillator were first
observed in photonic systems [Slu85] and extensively studied in [Bre97]. Additionally,
phononic [Mee96], number state [Orz01] and spin state [Est08] squeezing were observed
in ion systems and Bose-Einstein condensates. The properties of squeezed thermal states
are subject to theoretical studies since a long time [Fea88, Kim89, OV91, Wan93, Mar93a,
Mar93b]. Squeezed thermal noise has first been experimentally realized in a Josephson
parametric amplifier [Yur88]. However, while interactions of quantum systems with
squeezed thermal reservoirs have been analyzed theoretically [Kal06, Ban08, Dod11,
Sha13a], we only find a small number of applications to quantum optics [Kie15, Lo15],
while their use in quantum thermodynamics and their effect on the performance of
thermodynamic machines is largely unexplored to date.
In the following, we continue our investigations on the harmonic oscillator Otto cycle.
To analyze the effect of squeezing, we couple the engine to a high-temperature squeezed
thermal reservoir, while the low-temperature reservoir is still purely thermal. We find
that the efficiency at maximum power rises exponentially with the squeezing parameter
κ, surpassing the standard Carnot limit and converging towards unity exponentially.
We further present an experimental scheme to mimic the interaction with a squeezed
thermal reservoir for the single trapped ion system by combining reservoir and state
engineering techniques [Lei03]. Again, we demonstrate the experimental realizability
of such a scheme by Monte-Carlo simulations with realistic trap parameters and laser
interaction. We show that the single-ion engine may run at maximum power up to
an efficiency which is four times larger than the efficiency obtained with two thermal
reservoirs and a factor of two above the standard Carnot bound. In order to fulfill the
40
3.3. Employing non-thermal reservoirs
constraints of the second law, we deduce a generalized Carnot bound, which limits the
efficiency of an engine operating between squeezed thermal reservoirs.
3.3.1. Otto cycle with a squeezed thermal reservoir
We start with an analytical investigation of the system. Therefore, we consider a quantum
Otto cycle for a harmonic oscillator similar to Sec. 3.1.2, which is shown schematically
in the inset of Fig. 3.8 (a). The hot reservoir is replaced by a squeezed thermal reservoir
with a hot inverse temperature β2 and squeezing parameter κ. During interaction, the
harmonic oscillator relaxes to a non-displaced squeezed thermal state with mean phonon
number [Mar93a]
〈n∗(β,ω,κ)〉 = 〈n〉+ (2 〈n〉+ 1) · sinh2(κ), (3.26)
where 〈n〉 = 〈n(β2,ω2)〉 is themean thermal occupation number, see Eq. (3.4). We assume
the duration of this interaction to be much shorter than the duration of the isentropic
process, and thus it is assumed to take place at constant frequency ω2. Consequently, it
corresponds to an isochoric process. The mean energy after equilibration with a squeezed
thermal reservoir is given by [Mar93a],
( ) [ ( ) ]
〈H〉 = h̄ω〈n∗ 〉 h̄ω βh̄ω 1s (β,ω,κ) = coth 1+ 2+ 22 2 〈n〉 · sinh κ . (3.27)
In the following, we will keep the inverse temperature β2 constant and vary the
amount of squeezing κ, increasing the energy of the state C in the cycle. The additional
energy is still present at D, after the isentropic and adiabatic compression. The cycle is
closed by coupling the system to the cold thermal bath. Because of the stochastic nature
of this process, it erases any phase relation and thus thermalizes the squeezed thermal
state, resulting a pure thermal state with cold temperature β1 at point A. The mean
energy of the four stages of the Otto cycle now reads [Aba15]
( )
〈 〉 h̄ω β h̄ωH A = 1 coth 1 1 , (3.28a)2 ( 2 )
〈 〉 h̄ω β h̄ωH B =
2 coth 1 1 , (3.28b)
2 ( 2 )
〈 〉 h̄ωH = 2 β h̄ωC coth
2 2
∆H(κ), (3.28c)
2 ( 2 )
〈 〉 h̄ω1 βH = coth 2h̄ω2D ∆H(κ), (3.28d)2 2
with the non-thermal energy addition ∆H(κ) = 1+ (2+ 1/〈n〉) sinh2 κ.
Work is exchanged by the oscillator during the compression and expansion phases,
whereas heat and non-thermal energy are exchanged during the relaxation with the
reservoirs. Similar to the procedure in Sec. 3.1.2 we find the efficiency as ratio between
the net work of the engine and the total (non-)thermal energy supplied to the system,
−〈W1〉+ 〈W3〉 − ωη = 1〈 〉 = 1 , (3.29)Q2 ω2
which is unaffected by the squeezing (it cancels out) and equal to Eq. (3.15). This is an
important statement, since it shows that by employing squeezed thermal reservoirs, we
41
3. Results of analytical and numerical investigations
a) 1.0 b) 40
0.8
30
0.6
E C
rad 20
0.4 D
B
10
0.2
A ωrad
ω ω
1 2
0
0 1 2 3 4 0.0 0.5 1.0 1.5 2.0
Squeezing parameter κ Squeezing parameter κ
Figure 3.8. – (a) Efficiency at maximum power η∗ of the Otto engine, Eq. (3.18), plotted as a function
of the squeezing parameter r. Black, red and blue lines correspond to the temperature
ratios β2/β1 = 0.9, 0.6, and 0.3, respectively. The dashed lines in the corresponding color
denote the standard Carnot efficiency for each temperature ratio. The inset shows the energy-
frequency diagram of an idealized Otto cycle. Squeezing is applied during the hot bath
interaction between points B and C, leading to an increase of η∗ with κ, approaching unity
exponentially. (b) The corresponding maximum mean work 〈W〉max normalized to 1/β2 and
using a frequency ratio ω2/ω1 according to the optimality condition Eq. (3.31).
do not influence the intrinsic energy conversion process of the heat engine. However, the
squeezing provides additional energy to the system. Because of the constant η, the mean
work of the engine increases with increasing input energy,
( ) ( ) ( )
〈W〉ht(κ) = 〈 〉 〈 〉
1 ω2 − 1 ωW1 + W3 = 1 + 1 − 1 · 1+ 2 sinh2 κ , (3.30)β1 ω1 β2 ω2
where we used again the high temperature approximation. As a consequence of the
enhanced output power, the efficiency at maximum power η∗(κ) rises. To calculate η∗(κ),
we follow the procedure as above and maximize the mean work Eq. (3.30) with respect to
the frequency difference ∆ω = ω2 −ω1, while we keep all parameters but ω2 constant.
In the high-temperature limit, h̄βiωi ≪ 1, we find that the power is maximum when the
frequencies satisfy the condition
√
ω2 β1(1+ 2 sinh
2 κ)
= . (3.31)
ω1 β2
As a result, the efficiency at maximum power is given by
√
β
η∗(κ) = 1− 2 2 , (3.32)β1(1+ 2 sinh κ)
an expression which depends explicitly on the degree of squeezing κ. For thermal
reservoirs (κ = 0), this equation is equivalent to η∗ in Eq. (3.18). Remarkably, η∗(κ) rises
with increasing κ and approaches unity exponentially for large κ,
√
η∗
β
(κ ≫ 1) ≃ 1− 2 2 exp(−2κ). (3.33)
β1
42
Efficiency η*
Work Wmax (1/β2)
3.3. Employing non-thermal reservoirs
Fig. 3.8 shows the increasing η∗(κ) and the corresponding maximum work output
〈W〉max(κ) for different temperature ratios. While the Curzon-Ahlborn efficiency for
thermal systems is by definition smaller than the standard Carnot efficiency, which is
indicated by dashed lines in Fig. 3.8 (a), we observe that η∗(κ) may surpass this classical
limit even at moderate squeezing parameters. Thereby it demonstrates how non-thermal
reservoir interactions may be employed to reach a regime which is inaccessible for pure
thermal systems. However, following from the second law for this non-equilibrium
situation, the efficiency is bound by a generalized Carnot efficiency [Hua12, Aba14],
gen β
η = 1− 2C 2 . (3.34)β1(1+ 2 sinh (κ))
Note that the standard Carnot efficiency is an expression of the second law for one
particular configuration: two thermal reservoirs at two different temperatures. Eq. (3.34)
extends this result to a more general non-equilibrium setting that involves one thermal
and one non-thermal reservoir. Comparing Eq. (3.34) to the standard Carnot efficiency,
ηC = 1 − β2/β1, indicates that non-thermal reservoirs may be characterized by an
effective temperature Teff in order to obtain the familiar form of the laws of thermody-
namics [Rei02a, Aba14]. For a squeezed thermal reservoir we find
( )
Teff = T 22 2 1+ 2 sinh (κ) . (3.35)
Consequently, squeezing a thermal reservoir can be seen as an amplification of its thermal
energy, increasing its total energy depending on its temperature. When coupled to an
engine, the increase of the output power strongly depends on the initial temperature of
the reservoir, T2 = 1/(kBβ2), see Fig. 3.8 (b).
3.3.2. Numerical simulations
To support our analytical findings, we consider again the realistic proposal for an Otto
heat engine presented in the section above. We have performed semiclassical Monte-
Carlo simulations of the Otto cycle using the same methods as before. We assume trap
frequencies of ωr/2π = 3MHz and ωz/2π = 36 kHz at the position of the ion. The
angle of the electrodes is θ = 20◦ and the distance of the ion to the radial electrodes is
r0 = 1mm.
The coupling to the reservoirs is mimicked as before by a combination of the velocity
dependent scattering forces of a laser beam and electric field noise on the trapping field,
both acting on the radial degrees of freedom. In the presence of a hot squeezed thermal
reservoir, the radial state of the ion equilibrates to the temperature of the reservoir,
but is additionally squeezed during the interaction [OV91]. We mimic the coupling to
such a squeezed thermal reservoir by combining reservoir engineering (for the thermal
component) and state engineering (for the squeezed component) techniques. Squeezing
of the radial state of the ion is implemented by modulating the radial confining potential
at double the trap frequency [Jan92, Gal09]. In order to make sure that no work is done
on the axial degree of freedom, the squeezing operation is implemented in such a way
that the mean value of the potential energy is not changed after the operation. To this aim,
the radial trap frequency ω ′r is first increased to ωr = ωr + ∆ω for a quarter of a radial
oscillation period, t′ = 2π/(4ω′r). Then the frequency is lowered to ω′′r = ωr − ∆ω for
another quarter of a radial oscillation period, t′′ = 2π/(4ω′′r ), before it is returned to its
43
3. Results of analytical and numerical investigations
p
C
1.5 x ωr
p ω+Δω
D ω
1.4 x ω-Δω
t' t'' t
p
B'
x
1.3
B
p
1.2
x
A
2.8 2.9 3.0 3.1 3.2
Radial frequency ω / 2π (MHz)
r 
Figure 3.9. – Comparison of simulated Otto cycles of the single-ion heat engine, with two thermal
reservoirs (red) and with a hot squeezed thermal reservoir (blue). The corresponding phase-
space distributions of the states are sketched next to points A,B’,C and D, showing the
change in temperature of a thermal state (rotational symmetric Gaussian distribution) and
the squeezing of those states (ellipses). The interactions with the hot thermal bath (BB′)
and the squeezing operation (B′C) are performed sequentially to discriminate both effects
in their dynamics. The squeezing leads to a significant increase of the produced work. In
order to suppress effects of micromotion, a 5MHz Fourier low pass filter has been applied.
Inset: Squeezing operation applied between points B′ and C. The radial trap frequency ωr is
parametrically switched to higher and lower values.
initial value ωr (see inset in Fig. 3.9). The implications of this squeezing operation, as
well as alternative techniques, are discussed in Secs. 3.3.3 and 3.3.4.
We numerically simulate engine cycles with different ∆ω to achieve different squeezing
parameters κ. The thermodynamic Otto cycle of the engine is shown in Fig. 3.9, where a
cycle including a squeezed thermal bath (blue points) is compared to that employing
pure thermal reservoirs with the same temperature ratio (red points). The difference
in the area of the cycles corresponds to the increase in energy due to the squeezing
operation. The phase space distribution (x,p) is sketched for each of the states. In
order to analyze the influence of the thermal and the non-thermal part of the reservoir
interaction separately, we simulate the coupling to the squeezed thermal bath in two
consecutive steps: first by heating the radial state of the ion, followed by a squeezing
operation. Since the reservoir interaction is assumed to be negligibly short and thus to
occur at constant radial frequency, the order and the separation of the two processes has
no influence on the dynamics of the engine, but is chosen for a better understanding.
Regarding the performance of the engine, not the relaxation process itself, but only the
relaxed state matters [Aba15].
The resulting efficiencies η∗ are shown in Fig. 3.10, where we matched the frequency
ratios to the optimality condition Eq. (3.31). At a fixed ratio of the bath temperatures
of β2/β1 = 0.88 we attain good agreement with the theoretical prediction for η∗(κ),
Eq. (3.32), for different squeezing parameters in the range 0 < κ < 0.4. The maximum
44
Radial mean energy  -23H  (10  J)
3.3. Employing non-thermal reservoirs
0.4
efficiency at maximum power η*
standard Carnot limit ηC
generalized Carnot limit η genC
0.3
0.2
0.1
0
0 0.2 0.4 0.6
Squeezing Parameter κ
Figure 3.10. – Numerical simulation of the efficiency at maximum power η∗(κ), given by the gener-
alized Curzon-Ahlborn efficiency (3.32), as a function of the squeezing parameter κ (solid
red line), at a fixed temperature ratio of β2/β1 = 0.88. The region below the red dashed line
(yellow shaded) corresponds to possible efficiencies with purely thermal bath interactions,
bounded by the standard Carnot limit. The results of the Monte-Carlo simulations (black
dots) demonstrate with the given trap geometry that by squeezing the thermal state with
κ ∼ 0.4 the efficiency can be increased by a factor of four, which is two times higher than the
corresponding standard Carnot bound. The black dotted line shows the generalized Carnot
limit Eq. (3.34) for an engine interacting with a hot squeezed thermal bath.
ratio ω2/ω1 is limited by the opening angle θ of the funnel-shaped potential and by
a maximum amplitude a for the axial coherent oscillation. Considering realistic trap-
ping potentials, this amplitude is chosen to be smaller than a < 1mm [Vah09]. These
constraints limit the achievable squeezing parameters to κ < 0.5 at maximum power.
We observe that for a squeezing parameter of κ ≃ 0.4 the efficiency is quadrupled and
two times higher than the standard Carnot limit. To achieve an comparable efficiency
with pure thermal reservoirs, maintaining a maximized power output, an increase of the
temperature ratio by 70% would be needed, whereat the output power is still 35% lower
than for an engine which employs squeezing. These numbers underpin the power of
non-thermal reservoirs in the context of nanoscopic engines.
3.3.3. Generation of squeezing
Let us now focus on an experimental realization of a squeezed thermal reservoir interac-
tion for trapped ions. Thermal reservoirs at different temperatures are engineered via
Doppler cooling and tailored electrical noise on the trap electrodes [Tur00a]. Squeezing
of the ground-state wave function of a trapped ion was first demonstrated using resolved
sideband excitation on the second motional sideband [Mee96]. However, this approach
requires a system initially in the ground state and long interaction times. As described
above, to realize squeezing for any initial temperature is to modulate the harmonic po-
tential at double the trap frequency [Jan92, Gal09]. This leads to an elliptical deformation
of the phase space distribution of the thermal state (see Fig. 3.9), which is the signature
45
Efficiency η
3. Results of analytical and numerical investigations
2.0
a) 5 b)
1.5 4
3
1.0
2
0.5 κ = 0 (thermal state)
κ = 0.2 1
κ = 0.4
κ = 0.6
0.0 0
0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0
Phase (rad.) Squeezing parameter κ
Figure 3.11. – Comparison of the width of pure thermal and squeezed thermal states. (a) Width
of the projection of the phase space distribution for one full rotation in phase space, for a
thermal state and squeezed states with different squeezing parameters κ, normalized to the
width of a pure thermal distribution. Dashed lines give the according mean variance for one
full rotation. (b) Normalized mean width of the projection as a function of κ.
of a squeezed thermal state. In combination with the randomized thermal interaction,
this operation mimics a coupling to a squeezed thermal reservoir: the squeezed thermal
state after interaction corresponds by all means to a state we would expect by relaxation
to a fictitious squeezed thermal reservoir.
To our knowledge, such a scheme to generate squeezed states of motion has not
yet been implemented experimentally due to the high demands on the control of the
electrode voltages. However, in the recent years there has been huge technical progress
on the precise and fast control of trapping potentials, allowing for demanding transport
operations [Wal12b, Rus14] and the controlled compression of ion crystals [Ulm13]. An
instantaneous bang-bang control of the trapping field has been demonstrated in 2016 with
nanosecond switching, which should be appropriate for the generation of large squeezed
states [Alo16a].
However, a more sophisticated approach for the implementation of squeezed reservoirs
for trapped ions has been recently demonstrated experimentally, where a dissipative
relaxation from a ground state into a squeezed state of motion is realized [Cir93, Kie15].
However, this requires the ions to be in the ground state of motion and the timescale of
the relaxation process is not yet short enough to meet the demands for our heat engine
system. But, certainly, this is the right path to follow.
For the characterization of the resulting squeezed states, an analysis by side-band
spectroscopy would be the appropriate choice [Mee96, Zen95, Cir93], while the work
output leads to an increasing amplitude of the oscillation in the axial direction, which
may be analyzed on camera images or again by sideband spectroscopy [Wal12b].
3.3.4. Remarks on the generation of work and the notion of efficiency
Now, we have closer look on the proposed generation of squeezed states. By modulating
the confinement in radial direction, the second moment of its probability distribution (its
variance) is accelerated, while the first moment (its center of gravity) stays unchanged.
Therefore, an external energy source has to perform work on the distribution of the
ion, and thereby increases its total energy. This is a unitary operation with no entropy
change and thus no heat transfer. However, the external work is performed on the
46
Width of projection (norm.)
Mean width of projection (norm.)
3.4. Conclusion and Outlook
radial mode only (agent), while the axial mode (piston) stays unaffected. The probability
distribution of the axial degree of freedom and its higher moments are not changed.
Consequently, no direct work is performed on the piston by the squeezing operation2, but
only indirectly by the mode coupling mechanism of the engine. Note, that the coupling
mechanism between agent and piston is designed such that it only senses the change
in the mean variance between low and high energy states of the agent and translates
it into mechanical work of the piston. It is not sensitive to any phase information of
the squeezed state, since ωr ≫ ωz. As the mean variance of a squeezed thermal state is
higher than the one of a pure thermal state with same energy, see Fig. 3.11, it results in a
higher output power.
We end this section with a short note on the energy sources of reservoirs in thermody-
namics: as we have mentioned in Sec. 2.3, it is agreed that the energetic costs of preparing
and providing the reservoirs are not included in the efficiency balance. As a consequence
of this general agreement, only the energy of the state after relaxation with the reservoir
enters, and, in our case, the dynamics of the relaxation process and the way of how the
energy is provided does not enter the efficiency balance. Therefore, the work performed
on the radial state may be seen as the work required to generate the reservoir.
3.4. Conclusion and Outlook
In this chapter we have presented an analytical framework to describe the dynamics
of an Otto cycle for a harmonic oscillator, which is valid in the classical as well as
in the quantum regime. We have derived and analyzed the work and the efficiency
at maximum power. We have combined the analytical framework with a numerical
study of our proposed single-ion heat engine setup, which represents, in our case, a
classical harmonic oscillator in a very good approximation. We have demonstrated the
experimental feasibility of our approach and have shown that, indeed, the efficiency
at maximum power may be reached by our system for a wide range of parameters.
Furthermore, we have pointed out how our system can be extended to engineered
non-thermal reservoirs, and that the efficiency at maximum power can be enhanced
dramatically. The use of non-thermal reservoirs offers additional degrees of control and
manipulation, which may be exploited to increase the work generation.
Our system may operate as a heat pump by inverting the sense of the rotation of the
thermodynamic cycle. In this way, a temperature difference can be generated between
two reservoirs initially at the same temperature, transferring heat from a cold to a hot
reservoir. Such a nano refrigerator may be useful to cool various kinds of oscillators by
coupling it to their motional degrees of freedom. The transfer of our analytical framework
to quantum refrigerators has been done for thermal heat baths by Abah and Lutz [Aba16],
and for the squeezed thermal reservoir approach by Long and Liu [Lon15b], both in
close cooperation with us.
The following part of this thesis is dedicated to the experimental realization of the sin-
gle ion heat engine, operated in a classical regime with thermal reservoirs at temperatures
which are high as compared to the quanta of the harmonic oscillator.
2 The small radial frequency change seen in Fig. 3.9 is due to the axial motion z(t) of the ion during the
squeezing, and not to the squeezing itself. A similar effect is observed during the interaction with the
thermal bath BB’.
47

4 Setup of the experiment
The experimental setup of the heat engine system consists of the funnel-shaped ion trap
inside a vacuum chamber, an electronic supply system, as well as different laser systems
for ionizing calcium atoms and for cooling and manipulating the trapped ions. The
individual parts are presented and discussed in detail in following sections.
4.1. The funnel-shaped ion trap
The heart of this experiment is the funnel-shaped Paul-trap, see Sec. 2.4, which provides
the tapered trapping potential where the heat engine mechanism is based on. Since our
trap is the first of its kind, we decided to keep its geometry as simple as possible. We use
a macroscopic rod trap design, composed of two flat end-cap electrodes, four cylindrical
radio-frequency (rf) electrodes, as well as four cylindrical biasing electrodes. The trap
assembly is shown in Fig. 4.1. We chose an angle of φ = 10◦ between the rf-electrodes
and the trap axis. This angle provides a reasonable coupling between the radial and the
axial degrees of freedom, while it still guarantees stable trapping conditions. In order
to bring the rf-electrodes as close together as possible on the narrow side of the funnel,
electrodes with a small diameter are required. We use commercial syringe-needles3
with an outer diameter of only 0.8mm as electrodes. The needles are made of stainless
steel, they provide a high manufacturing quality and are perfectly straight over a length
of several centimeters. Further discussion about the installation of these electrodes is
provided in App. A.1. At th√e narrow side of the funnel geometry, the electrodes are
mounted with a distance of 2mm to trap axis.
As end-caps we use 25× 25mm plates of machinable glass-ceramics4 with a thickness
of 4mm. These plates also serve as mounts for the trap-electrodes and thus facilitate the
construction of the trap, see Fig. 4.2. The inner faces of the ceramic plates are coated with
a nickel-molybdenum bonding-layer (150 nm) and a top-layer of gold (100 nm)5. This
coating provides an electrically conductive surface with little surface reactivity [Sch09].
The distance between the end-caps is chosen to be 8mm, which is a compromise between
a small geometry, a solid construction, and undisturbed electric field lines of the radial
trapping field at the center of the trap. Additionally, the end-caps have a hole in their
3 Sterican 21G cannula, Braun Melsungen AG.
4 Machinable Glass Ceramic MACOR, Corning Inc.
5 The bonding layer as been evaporated by the group of Prof. W. Heil. The gold layer has been applied by the
group of Prof. M. Kläui. We thank both groups for their cooperation
49
4. Setup of the experiment
Figure 4.1. – Images of the funnel-shaped ion trap. a) Trap after assembling, composed of an isolating
ceramic mount (A), four outer-electrodes (B), two gold-coated ceramic end-caps (C) and the
four rf-electrodes in funnel geometry (D). b) Top flange of the vacuum chamber with mounted
trap (A), multi-pin static voltage feed-through (B), the four rf-feed-throughs (C), the calcium
oven in its holder (D), a mechanical shutter for the atom beam (E) and an optical access for the
laser beam in vertical directions, which addresses the radial degrees of freedom of the ion (F).
center to give access to a laser beam propagating along the trap axis. The diameter of
the holes is 1.0mm, which allows for using a laser-beam with a homogenous intensity
profile over the center region of the trap without causing stray light from the electrodes.
Again, stray light of the laser beams may lead to charges on ceramic surfaces, which
distort the trapping potential. To prevent charging processes, a small copper tube with a
length of 2mm is inserted to the center holes on the trap-side and electrically connected
to the coating. The holes constrain the minimum distance of the rf-electrodes from the
trap axis to be no smaller than 1mm and thus restrict the geometry of the trap to the
dimensions chosen for our design.
To compensate for external electric stray fields and thereby to minimize micromotion,
an additional set of four cylindrical biasing electrodes complements the trap setup [Ber98].
The electrodes have a linear and parallel geometry and are placed behind the four rf-
electrodes, as shown in Fig. 4.2. To provide a rigid geometry we chose a diameter of
the outer-electrodes of 6mm and a distance of 10mm to the trap axis. Additionally,
since the diameter of these electrodes is much larger than the one of the rf-electrodes,
they provide a strong effect on the electric field at the center of the trap and only small
voltages are needed to move the ion to the node of the rf-field. This feature becomes
important when heating the ions with electrical noise via these outer electrodes, since
low noise amplitudes are sufficient to heat the ion considerably. To assure an electrical
isolation between the electrodes and the end-caps, the gold coating has been removed on
a 0.1mm band around the mounting holes of all electrodes.
4.1.1. The radio-frequency supply
In order to trap ions at reasonable trap frequencies, rf-signals with several hundred Volts
amplitude on the trap electrodes are needed. The rf-signal is generated by an analogue
signal generator6 and amplified by a 2W amplifier7. A helical resonator is employed
6 Signal Generator SMS2, Rohde & Schwarz GmbH & Co. KG.
7 ZHL-1-2W, Mini-Circuits. Water cooled to reduce temperature fluctuations.
50
4.1. The funnel-shaped ion trap
Figure 4.2. – Exploded-view drawing of the ion trap setup. The trap electrodes are mounted between
the two gold-coated end-caps, which are held by the outer electrodes. The whole trap setup
is mounted on a ceramic holder. The cutouts in the end-caps allow the optical access to the
center of the trap for the individual laser beams along diagonal directions.
for impedance matching between the 50Ohm rf-source and the high impedance ion-
trap [Mac59, Tol14]. The helical resonator is essentially a shielded LC circuit. A copper
coil, placed inside a metal cylinder, represents the inductance as well as a cylindrical
capacitor. The impedance of the ion trap, the cables and the vacuum feed-throughs
provide the load impedance and thus contribute to the resonance frequency of the entire
system [Siv12]. The rf-electrodes dominate the load impedance and provide a capacitive
load with Ct = 3.9 pF for each of the electrodes, and with negligible resistance and
inductance. The absolute value of the impedance for a purely capacitive load reads
|Zt| = [Ω −1rf C] . Consequently, for typical drive frequencies of Ωrf/2π ≈ 20MHz the
trap impedance |Zt| ≈ 105Ohm is much larger than the impedance of the rf-source,
|Zs| = 50Ohm. For this reason, a variable impedance matching is indispensable.
As mentioned in Sec. 2.4.3, we use an anti-symmetric rf-drive signal fed to the four
trap electrodes. Such a signal is obtained by employing a helical resonator in λ/2-
geometry, where the coil has a ground connection at its center and the two open ends
are connected to the trap electrodes. Fig. 4.3 (a) shows a cutaway-model of our helical
resonator design. The rf-signal is coupled to the coil via a variable sliding-contact. The
resulting oscillating voltage inside the resonator system has a node at the center and
two anti-nodes at the open ends. Variable capacitors at the output of the resonator
allow for a fine adjustment of the phase and the amplitude ratio of the output signal,
see Fig. 4.3 (b). Additionally, two capacitive dividers on the resonator outputs allow to
monitor an rf-signal with reduced amplitude. In our setup, we employ two pieces of
BNC-cables with different lengths (RG-58 with a capacity of 68 pF/m) to realize a divider
with a little total load capacity. The division factor is then defined by the length ratio
of the two pieces and has been measured to be κ = 118. The recorded monitor signal
is presented in Fig. 4.4, with a resulting resonance frequency of the entire oscillating
circuit of Ωrf/2π = 21.41MHz. Since Ωrf is large compared to all trap frequencies,
crosstalk and resonance effects are avoided. A high Ωrf reduces the amplitude of the
residual micromotion, but requires higher amplitudes of the driving signal to achieve
the desired trap frequency ω [Lei03]. With this setup, we reach amplitudes of VΩ ≈ 1 kV
51
4. Setup of the experiment
a) to trap to trap
input
b) RF +rf
Oszi
S
_
SC 0V +
VA _+
A
-rf
helical resonator trap
Figure 4.3. – The helical resonator in λ/2 geometry. (a) Cutaway-drawing of the silver-coated
housing and the helical resonator (HR) with mass contact (magenta) at the center of the copper
coil. A threaded Teflon cylinder (yellow) is screwed inside the coil; it moves the sliding
contact (blue) along the coil and allows for precise impedance matching. (b) Equivalent
circuit diagram of the HR setup. The rf-signal at 21.41MHz is provided by a signal generator
(RF). A switch (S) and a variable attenuator (VA) allow to manipulate the trap potential. The
rf-signal is amplified by a 2W amplifier (A) and fed to the inductivity of the HR by the sliding
contact (SC). The HR has additional variable capacities on both output channels to adjust for
deviations of the rf-signal on the trap electrodes. Capacitive dividers allow to monitor the
rf-signal on a 2-channel oscilloscope (Oszi). The two output signals with opposite sign drive
the capacitive load of the 4 trap electrodes in a diagonal configuration.
peak-to-peak on the trap electrodes, resulting in secular frequencies of the trapped ion in
the radial directions of ωx,y/2π ≈ 500 kHz. A switch8 and a variable attenuator9 allow
for clearing the trap and for controlling the depth of the trapping potential precisely.
While a perfect anti-symmetric trap drive prevents the ion from any micromotion in
axial direction, a phase shift and an amplitude mismatch between the two signals causes
residual micromotion. From the monitor signal in Fig. 4.4 (a) we deduce a phase shift
between the two outputs of ∆φ = 179.74(6)◦, thus a resulting phase error of δφ < 0.3◦.
The relative amplitude difference is measured to be ∆VΩ/VΩ = 4.8%. Both effects add
up to an excess axial micromotion, depending on the displacement d of the ion from the
node of the rf-field. For the given parameters and an axial displacement of d = 50 µm,
the micromotion has been simulated be provide an amplitude of a < 50 nm. Its relative
amplitude of a/d < 0.1% is negligible for all applications within this thesis. Note that not
8 ZYSWA-2-50DR, Mini-Circuits.
9 ZX73-2500-S+, Mini-Circuits.
52
4.1. The funnel-shaped ion trap
Figure 4.4. – (a) Opposing radio frequency signal on the two output lines of the helical resonator,
measured via the capacitive dividers andmultiplied by κ. Fitted sine-functions reveal a relative
phase of ∆φ = 179.74(6)◦ between the two signals and a relative amplitude difference of
∆A/A = 0.05. (b) Spectrum of the normalized power absorption of the resonance circuit. The
full width at half maximum ∆Ω = 92(1) kHz reveals a Q-factor Q = 237(3). (c) Impedance of
the resonance circuit as a function of the probe frequency. The resonator is tuned to match the
50Ohm impedance of the input signal at resonance.
using an anti-symmetric trap drive would result in a relative micromotion of a/d = 6%
for a similar displacement, see [Tol14].
The measured spectrum of the power absorption of the helical resonator system is
shown in Fig. 4.4 (b). The fit of a resonance curve reveals a quality factor of the entire
resonator system of Q = Ωrf/∆Ωrf = 237(3) [Dem08]. Therefore, the resonator provides
a good filtering of the input signal and suppresses electric noise as well as higher orders
of the drive frequency efficiently, see [Tol14]. The quality factor corresponds to the
amplification of the input signal by the resonator, VΩ = QVin, and allows to provide rf-
signals with high amplitudes VΩ to the trap electrodes. The effect of the helical resonator
on the impedance of the entire resonator system is shown by a measured spectrum in
Fig. 4.4 (c). At the resonance frequency, the impedance of the system matches 50Ohm
and thus the reflected part of the input signal is strongly reduced.
4.1.2. The constant voltage supply
Ambitious experiments with trapped ions impose high demands on the constant voltages
applied to the trap electrodes, since any electrical noise leads to artificial and undesirable
heating-rates [Bro15]. In our experiment, the end-caps and the biasing-electrodes are
supplied with constant voltages from a 12-channel low noise DAC (Digital-to-Analog
Converter) unit, which has been developed in our group10, see [Zie12]. It provides an
actively stabilized voltage in the range of -10V to +10V with 16-Bit resolution and a noise
level below 20 µV. Typical voltages on the end-caps are 7.0V on the wide side of the taper
and 9.0V on the narrow side, leading to an axial trap frequency of ωz/2π = 170 kHz.
In order to reduce the excess micromotion, the outer-electrodes are biased with typical
voltages between 0.9V and 3.5V. Furthermore, two opposing outer-electrodes have an
additional offset of 1.5V in order to break the symmetry of the radial trapping potential
and to lift the degeneracy of ωx and ωy. The individual DAC channels are controlled by
10 Op40 DC Segment Controller, H. Lenk, University of Mainz.
53
4. Setup of the experiment
Figure 4.5. – Electronic setup of the noise inversion circuit. The noise signal is provided to a balance-
unbalance transformer with an equal turns ratio and a low frequency cut-off far below the
switching frequency. To eliminate high frequency terms, we use an additional low pass filter
(LP). The resulting signals on both output lines of the secondary coil have the same amplitude
but inverted sign. A bias-tee in both lines allows to superpose the noise signal with the static
voltage (DC), which is used for compensating micromotion. Variable resistors allow to adjust
for differences in the amplitudes on both lines.
a field-programmable gate array11 (FPGA), allowing for fast and precise voltage changes
at a clock frequency of 50MHz. Additionally, the FPGA controls a 30-channel TTL-unit,
which triggers all time-relevant components of the experiment.
4.1.3. Electric field noise
As we will shown in Sec. 6.1.2, the most efficient way for controlled heating is to expose
the trapped ion to a combination of white electric field-noise and laser cooling. A
Gaussian probability distribution of the noise signal then guarantees the generation of
hot thermal states of the ion. In our setup, electrical noise is provided to two opposing
outer-electrodes, thus effecting the radial component of the trapping field.
However, by applying the electric field noise to the electrodes, we want to excite
a thermal Brownian motion of the trapping potential without changing the external
parameters like the trap frequencies. Therefore, we need to superpose the trapping
potential with a position independent electric noise field. This is reached by applying
the same noise signal with inverted sign on two opposing outer-electrodes. In the
experimental setup, we achieve a differential noise signal by feeding the noise to a
balun transformer12, as shown in Fig. 4.5. White and Gaussian voltage noise causes
naturally a white and Gaussian current noise in the primary and secondary windings
of the transformer. When identical loads are connected to each side of the secondary
windings, the current in both loads has the same absolute value but inverted sign. As a
result, the voltages on the two outer-electrodes, being capacitive loads, have the same
amplitude but inverted sign as well. A measurement of the two noise signals is presented
in Fig. 4.6, together with the sum of both. For the given exemplary voltage signals,
the residual standard deviation of the summed signal increases by 0.014V above the
background signal when noise is applied. This increase leads to a fluctuation of the
radial trap frequencies, but is negligible as compared to the standard deviation of the
noise signals, 0.3V, and even more as compared to the trapping fields VΩ. Consequently,
applying the same noise signal with inverted sign to opposing outer electrodes results in
11 Avnet Virtex-5 FXT Evaluation Kit, Avnet electronics marketing, Phoenix, USA.
12 FTB-1-6*A 15+, Mini-circuits.
54
4.1. The funnel-shaped ion trap
Figure 4.6. –Noise signal which is fed to the electrodes. (a) and (b) Original and inverted noise
signal after the inversion circuit. As it is needed for the heat engine operation, the noise is
switched on and off with a repetition rate of 82 kHz and a duty cycle of 0.3. (c) The sum of
the original and the inverted noise signal. The two signals cancel each other to a remaining
standard deviation by 0.014V above the background signal.
a random movement of the trap potential with negligible effect on the trap frequencies.
4.1.4. Loading ions into the trap
The 40Ca+-ions are loaded into the trap by ionizing neutral calcium atoms within the
trap volume. Therefore, we use a resonant two step photo-ionization process. The
neutral atoms are thermally evaporated from a hot cavity. We resistively heat the rear
part of a titanium tube, filled with granular calcium, with a typical current of 5.2A. The
effusing atoms are collimated and directed by the extended geometry of the oven tube
in order to pass the volume of the trap. Inside the trap, the atoms are first resonantly
excited and then non-resonantly ionized [Ebl11, Tol14]. As soon as the Ca-atoms are
ionized, they are interacting with the (oscillating) electric fields and confined within the
trapping potential, as long as their thermal energy is lower than the depth of the potential,
typically Ekin . 5 eV. Subsequently, the ions are Doppler cooled on the S1/2 ↔ P1/2
transition and form crystalline structures at the minimum of the trapping potential due
to Coulomb interaction. The increasing Coulomb energy for each additional ion allows
to reduce the number of ions deliberately by briefly lowering the potential depth below
the corresponding threshold energy, such that, for instance, only one ion remains.
4.1.5. Detection of fluorescence light
The fluorescence light of the trapped ions at 397 nm is collected by a lens system with a
numerical aperture of 0.26, see [Ebl11]. The image of the ions is projected onto the photo-
55
4. Setup of the experiment
Figure 4.7. – Camera image of a linear 5-ion crystal, taken by the EMCCD camera with an exposure
time of 100ms. The corresponding trap frequencies are 170 kHz axial and 420 kHz radial. The
distance of the ions is evaluated via the effective pixel size of 0.667 µm and results to 18 µm
between the neighboring inner ions and 20 µm between the the outer ions and their neighbors.
These values agree with the simulation of an ion crystal at the same trap frequencies.
sensitive chip of a detector. Within this thesis three different devices are employed to
detect the fluorescence light of the ion: an EMCCD13-camera, an ICCD14-camera, as well
as a photomultiplier tube15 (PMT). For detecting the ions we operate the EMCCD-camera
with a chip size of (8× 8)mm and (1002× 1004) pixels, providing a quantum-efficiency
of 0.4 at 397 nm. For the imaging system, a magnification factor of 12.0(1) is obtained
from a comparison of the measured ion distance on camera images and the calculated
ion positions from simulations with the given trap frequencies. It leads to a spatial
resolution of 0.667(5) µm-per-pixel on the camera images, which is sufficient for a precise
determination of the ion positions, see Fig. 4.7. An optical bandpass filter16 reduces the
background light at wavelengths different than 397 nm by 10−4. The exposure time of the
camera is limited by internal electronic noise originating from thermal excitations within
the potential wells of the individual pixels, as well as due to the shift operations during
the readout process [Ise15]. These dark-counts compete with the fluorescence-signal of
the ion, limited by its scattering rate, the numerical aperture of the objective, as well as
the quantum-efficiency of the camera. Using this camera setup, an exposure time of 1ms
is necessary to detect an ion at all, while for a precise determination of the position of the
ion the exposure time should exceed 50ms. This time is long compared to the timescales
of dynamic processes of the ion, such as secular frequencies or thermalization processes.
In order to circumvent the issue of limited exposure time we have used the PMT and the
ICCD-camera for time-critical measurements.
The PMT is well suited for all measurements where short exposure times are crucial,
but where no spatial resolution is needed. It provides a quantum efficiency of 0.27
and almost no dark-counts. It records single photons and thus enables us to measure
with exposure times of only some microseconds. The PMT is operated in parallel to
the camera. A beam-splitter between the objective and the camera reflects 30% of the
fluorescence-light onto the photo-cathode of the PMT. An adjustable aperture in the focus
of this beam path suppresses most of the background light. The single-photon events are
counted and processed by a self-programmed 50MHz micro-controller17 (MC). For more
details, see App. A.2. As we demonstrate in Sec. 6.6, we have successfully reduced the
exposure time to 5 µs.
13 Electron Multiplying Charge-Coupled Device, iXon 885, Andor Technologies Ltd.
14 Intensified Charge-Coupled Device, PI-MAX4:1024EMB, Princeton-Instruments.
15 P25PC, Electron Tubes Enterprises Ltd. (now being sold by Sens-Tech Ltd.)
16 BP 394nm x 10nm OD4, Edmund Optics Ltd.
17 32 Bit and 50MHz, PIC32MX250F128B, Microchip Technology Inc.
56
4.1. The funnel-shaped ion trap
Figure 4.8. – Acquisition process of the ICCD camera. (a) Single fluorescence photons from the ion
are collected by an objective lens (L) and converted into single electrons by a photo cathode
(PC). The electrons are accelerated towards amulti channel plate (MCP) by a control voltageUa
and multiplied. The resulting electron bunches are accelerated by a constant voltageUb onto a
phosphorus screen (PS), where they are back converted into photon bunches and recorded by
a CCD camera. The gain of the device is controlled by the MCP voltage Ui . (b) Reconstruction
of the ion trajectory. The fluorescence distribution from images integrated on the CCD chip is
projected on the trap axis (blue points) and fitted by a normal-distribution (blue line). This
precisely reveals the center position of the thermal distribution of the ion. Taking those images
at different phases of the oscillation allows to reconstruct the ion trajectory (orange points and
line).
To resolve the oscillation of a trapped ion we need both, a time resolution in the
microsecond range, as well as a spatial resolution on the micrometer scale. These
requirements are complied by the ICCD camera, which combines the advantages of a
PMT and a CCD camera. Fig. 4.8 illustrates the acquisition process: the fluorescence
photons are converted to electrons by a photo-cathode and accelerated by a voltage
difference towards a micro-channel-plate (MCP), where the electrons are multiplied. A
subsequent phosphorus-screen converts the electrons back to photons which are detected
by a CCD camera. Switching the photo-cathode voltage to a level higher than the
MCP voltage, the photo-electrons are repelled by the MCP and detection is avoided.
The camera chip provides (1024× 1024) pixels at a size of (13× 13) µm, leading to an
effective pixel length of 1.07(9) µm on recorded images. The camera system shows a total
quantum efficiency of 0.1 at 397 nm. The gating mechanism allows for exposure times
as short as 2 ns. However, to record a resolved oscillation of the ion, we use a typical
exposure time of 700 ns. The internal gating is triggered externally and synchronized
with the driven oscillation of the ion, and the fluorescence at a fixed phase φ of the
oscillation is integrated for typically 105 frames on the CCD-chip. The phase φ is then
scanned over a full oscillation period and an analysis of the ion position for each φ allows
to reconstruct the resolved oscillation of the ion [Näg98]. A sinusoidal fit of the data
reveals precisely the amplitude of the oscillation and its phase relative to the driving
signal. The ICCD camera was a loan from Princeton-Instruments and was to our disposal
only for a three weeks period18.
18 We thank Princton-Instruments and especially Norbert Gulde for the collaboration.
57
4. Setup of the experiment
4.1.6. Generation of a magnetic quantization field
To apply a static magnetic field in order to provide a defined quantization axis and to
lift the degeneracy of the fine structure levels of the Ca-ions, we have mounted two
orthogonally oriented coil-pairs in the horizontal plane, as well as one coil for the vertical
direction. The coils for the horizontal magnetic field components have each a diameter of
120mm and 210 windings. They have a distance of 170mm to the center of the trap and
are mounted symmetrically on the flanges of the vacuum chamber. Each opposing coil
pair is operated in a quasi-Helmholtz configuration, such that the magnetic field can be
assumed to be homogenous over the length of the ion trap. The vertical magnetic field
component is generated by a single coil with a diameter of 240mm and 280 windings,
which is placed on the top flange at a distance of 115mm to the center of the trap, again
leading to a nearly homogenous magnetic field inside the trap volume. The individual
magnetic field strengths are tuned to compensate for external magnetic fields (mainly the
earth magnetic field) and to provide a well defined quantization field of B = 4.7× 10−4 T
in the horizontal plane. This results in a Zeeman-splitting of the atomic sub-states along
the direction of ~B. At the given magnetic field strength B, the ground state Zeeman-
splitting equals ∆ωZ/2π = 13.2MHz. By choosing the appropriate polarization of the
laser beams we can address the sub-states individually. To facilitate the addressing of
the individual spin-states of the ion, the direction of the magnetic quantization field is
oriented along one of the optical axes of the vacuum-chamber and has an angle of 45◦to
the trap axis.
4.2. The vacuum setup
In order to reduce the probability of collisions of the trapped ion with background gas
particles, we operate the trap setup in an ultra high vacuum (UHV) environment. We use
a stainless steel vacuum vessel with a volume of 1.4× 10−2m3, which is copper-sealed
against the environment. Employing a titanium-sublimation pump19 and an ion-getter
pump20 simultaneously, we reach a final pressure of P −11UHV < 2× 10 mbar. With such
a low pressure, the collision rate sr of the trapped ion with background gas particles is
reduced to s < 10−6r s−1.
The trap is mounted on the CF200 top flange of the vacuum vessel, as shown in Fig. 4.1.
The optical access of laser beams in the horizontal plane, as well as the imaging of the
trapped ions, is allowed by 8 CF60-viewports. For the camera objective, an inverted
viewport is installed, which allows to place the objective at a distance of 61mm to the
trap center [Ebl11]. In order to provide a laser-cooling beam in the vertical direction,
an additional viewport is installed on the top flange. On the bottom side we use a
combination of an in-vacuum mirror with a viewport in the sidewall of the vessel, which
becomes necessary by the position and geometry of the vacuum pumps.
4.3. The laser system
For ion trap experiments laser light at different wavelengths is needed: for the ionization
of Ca-atoms, and for cooling, manipulating, as well as for detecting the trapped Ca-ions.
19 TSP, tectra GmbH.
20 VacIon Plus 75 StarCell, Agilent Technologies.
58
4.3. The laser system
As shown in Sec. 2.7, the relevant part of the energy level structure of 40Ca+-ions can
be reduced to a three-level system with fine-structure levels 2S1/2, 2D3/2 and 2P1/2. The
cooling transition S1/2 ↔ P1/2 is addressed by a lightfield near 397 nm. In order to
reach a closed cycling transition for effective laser cooling, all the population which
decays into the metastable state D3/2 is pumped back to the P1/2 state by a laser at
866 nm. A controlled and efficient excitation of internal states of the ion requires laser-
light sources which are stable in frequency and intensity. We employ grating-stabilized
diode laser systems, which meet all requirements for our experiments: they are available
at the desired wavelengths, they run in single-mode operation with a typical spectral
linewidth below 1MHz (measured in Sec. 6.3), they provide an optical output power on
the order of 10−2W, and show a long durability at low maintenance effort. To obtain the
demanded frequency stability, the lasers are locked to an external cavity. Fine adjustment
of frequency and power, as well as switching of the laser beams, is achieved by acousto-
optic modulators (AOM). The wavelengths of all lasers are continuously measured by
an 8-channel wavelength-meter21 with an accuracy of 30MHz. In the following we will
discuss the individual elements of the laser setup.
4.3.1. Laser system for ionizing neutral Ca-atoms
The ionization process of neutral 40Ca-atoms consists of two excitation steps: first, a reso-
nant excitation from the ground-state [Ar] 4s2 1S 10 to the excited-state [Ar] 4s 4p P1, fol-
lowed by a non-resonant ionization with a photon energy exceeding the ionization poten-
tial (IP). The resonant transition has a a total width ΓI = 35.0MHz at a transition energy
of EI = 23652.304 cm−1, corresponding to a wavelength of λI = 422.79230 nm22 [Kra15].
In order to drive this transition efficiently, we use a commercial extended-cavity diode
laser23 with a grating in Littrow-configuration, which provides a linewidth of about
1MHz. An optical power of 2mW is largely sufficient to saturate the resonant transition
rate of AI = 2.2× 108 s−1 [Kra15]. The resonance wavelength has to account for the
Doppler shift of the thermal atoms, depending on the relative direction between the
atom beam and the laser beam.
For the non-resonant ionization step, the electron in the excited state absorbs a photon
with an energy high enough to overcome the atomic binding energy. Therefore, the
photon energy has to be larger than the difference between the excited P1-state and the
IP at 49306.3 cm−1 [Kra15]. The non-resonant ionization cross-section, typically of the
order of 10−17 cm2 [Rae10], is low compared to resonant transitions and decreases with
increasing photon energy. As a consequence, this process limits the efficiency of the
ionization and thus requires a high photon flux density and an energy above, but close to
the IP. Therefore, we use a free-running laser diode with a central wavelength at 375 nm
and a total output power of 8mW.
The two laser beams are superposed with a polarizing beam splitter and focused into
the trap volume. We achieve typical rates for ionization and subsequent trapping of
10 s−1, depending on the atom flux which is set by the temperature of the atom source.
Mechanical shutters in both laser beam paths, built from modified loud-speakers [Sin02],
allow for controlled loading of a desired number of ions.
21 WS-U 30, HighFinesse GmbH.
22 In this thesis, all optical wavelengths are given as vacuum wavelengths.
23 DL 100, TOPTICA Photonics AG.
59
4. Setup of the experiment
Figure 4.9. – Technical scheme of the laser setup for cooling and detecting the ions. The light emitted
from the laser diode at 793 nm is amplified in a temperature stabilized tapered amplifier (TA).
Subsequent second harmonic generation (SHG) in a Lithium-Triborate (LBO) crystal, placed
in a ring cavity, generates a lightfield at 397 nm. Typical laser power on each step is noted.
One of the mirrors of the SHG-cavity is mounted on a piezo, where a Pound-Drever-Hall
(PDH) locking scheme regulates the cavity length and compensates for thermal drifts. To
level out fast frequency fluctuations, the PDH-error-signal is fed to the field-effect transistor
(FET) of the current modulation of the laser diode via the dc-input of an internal bias-tee. The
397 nm beam is distributed on different pathways via half-wave plates (λ/2) and polarizing
beams splitters (PBS). Each beam provides an acousto-optic modulator (AOM) in double pass
configuration and is guided towards the experiment via single mode fibers (SMF). The second
laser diode runs at 866nm. Its beam passes through an AOM in double pass configuration,
which allows for frequency scans in a range of 150MHz without changing the beam pointing.
Two different VCOs connected to the AOM allow for fast and precise frequency jumps,
triggered by the field-programmable gate array (FPGA). Both lasers are locked to an external
cavity, see Fig. 4.11.
4.3.2. Laser systems for cooling and detecting the ions
The S +1/2 ↔ P1/2 dipole transition of the Ca -ions is used for laser cooling and provides a
scattering rate of 1.4× 108 s−1 [Kra15], resulting in a natural linewidth of ΓSP = 22.1MHz,
see Sec. 2.7. To drive this transition we use a grating-stabilized diode laser in Littrow-
configuration24 at 793.91845 nm. The laser beam is first amplified by a tapered amplifier
(TA) and subsequently frequency-doubled in a resonant cavity (Second Harmonic Gener-
ation, SHG) to obtain photons at 396.95923 nm. The setup of this laser is shown in Fig. 4.9.
In order to keep the output power and the frequency on a constant level, the wavelength
of the laser diode and the length of the SHG-cavity are stabilized on each other, using the
Pound-Drever-Hall (PDH) locking technique [Bla01], see Fig. 4.9. As re-pumping laser,
driving the D3/2 ↔ P1/2 transition, we use a home built grating-stabilized diode laser in
Littrow-configuration, running at 866.45090 nm.
In a standard configuration, the laser beams at 397 nm and 866 nm are oriented parallel
to one another and perpendicular to the magnetic field, see Fig. 4.10. They have an angle
of 45◦ to the trap axis and 60◦ to the radial directions, and thus all vibrational modes
are addressed simultaneously. The polarization of both laser beams is adjusted to be
linear and perpendicular to the magnetic field, leading to a superposition of σ+ and
24 DL-SHG 100, TOPTICA Photonics AG.
60
4.3. The laser system
Figure 4.10. – Orientation of the individual laser beams with respect to the ion trap, in a top-view
(left) and a side-view (right). The capital letters correspond to the output beams of Fig. 4.9.
While the 866 nm beam is always used in a diagonal orientation, the 397 nm beam has
different possible directions, to address and manipulate the states of the ions selectively. The
fluorescence light is captured by an objective and projected on a camera (EMCCD or ICCD),
as well as on the PMT. The data of both devices is read out and processed by the control
computer (PC) and represents the main observable in our experiments.
σ− polarization with respect to the magnetic quantization axis. This configuration is
advantageous for the resolution of dark resonances, see Sec. 6.2. The intensity and the
size of both laser beams at the position of the ions are typically 35 µW at a beam waist of
160 µm for the 397 nm laser and 80 µW at a beam waist of 380 µm for the 866 nm laser,
saturating the individual transitions. A flip-mirror permits to change the beam path of
the 397 nm laser to propagate parallel to the direction of the magnetic field (beam path B’,
dashed line in Fig. 4.10). The polarization of this beam path is again chosen to be linear,
such that both σ+ and σ− transitions are driven. An additional beam of the 397 nm laser
is oriented along the axis of the trap and addresses the axial degree of freedom of the
trapped ions only. Finally, a 397 nm beam crosses the trap volume in vertical direction
and has an angle of 45◦ to both radial direction, x and y. It addresses the radial states
of motion exclusively, and will be used to control and change the temperature of these
degrees of freedom. In order to avoid a radiation pressure induced force on the ion in
vertical direction, we apply two counter-propagating beams25, as shown in Fig. 4.10.
4.3.3. Scanning and switching of the beams
Acousto-optical modulators26 (AOM) are installed in all beam-paths of both lasers.
Double pass configurations are used to control the intensity of the beams and to tune
25 This doesn’t lead to a standing light wave, since the beams are detuned by 2MHz and sent through a 5m
single mode fiber each, whereby thermal fluctuations destroy the relative coherence between both beams.
26 397 nm setup: QZF-80-20, Brimrose Corp.; 866 nm setup: TEF-270-100, Brimrose Corp.
61
4. Setup of the experiment
Figure 4.11. – The reference cavity is composed of two mirrors in hemispherical configuration (yel-
low), mounted on a spacer (blue) and is placed in a vacuum chamber to prevent fluctuations
of its optical length. One of the mirrors is mounted on a stack of two counter-acting piezo-
rings (green) to tune the length of the cavity by a variable high-voltage (HV) supply. A
CCD-camera and a PD monitor the transmission signal of the cavity modes. The PDH-lock
is composed of a local oscillator (LO), a PD and a frequency mixer. The errorsignal is fed in
parallel to a PI and a PID circuit, which regulate the angle of the grating by a piezo and the
current of the laser diode by a field-effect transistor (FET).
their frequency without changing the beam pointing [Don05]. Each AOM is controlled by
an amplified signal of a home-built voltage controlled oscillator (VCO) system27, which
is adjustable in frequency and intensity, see App. A.4. An rf-switch allows for alternating
between two different VCO systems at different drive-frequencies on a timescale of 10 ns,
and is connected to the AOM in the branch of the 866 nm laser, see Fig. 4.9 and [Tol14].
To perform frequency scans of the fluorescence spectrum a constant intensity of the
AOM output over a range of 40MHz is required. However, an AOM has an RLC
resonance circuit for impedance matching, which shows a strongly frequency dependent
absorption profile. Here, the beam intensity is kept constant using a gauge function
for corrected driving power values as a function of the driving frequency, which was
measured previously with a photo-diode (PD). The power thus remains constant with a
stability > 99% during frequency scans, see App. A.5 for details.
4.3.4. Frequency stabilization
For cooling the trapped ions, the frequencies of both lasers are red-detuned with respect
to the corresponding atomic transition ∆397,866 < 0. The temperature of the ions during
laser-cooling strongly depends on the detuning ∆397 [Ste86a]. In order to achieve a
constant and well defined temperature of the ions, the frequency fluctuation of the
lasers should be reduced to a minimum. Furthermore, later in Sec. 6.4, we will exploit
narrow band resonances within the fluorescence spectrum of the ions to determine
their temperature, which may be washed out by frequency fluctuations of the lasers.
As a consequence, these fluctuations should be small as compared to the linewidth of
the corresponding atomic resonances. In order to reduce fluctuations and drifts, the
frequency of the diode lasers is stabilized on an external reference cavity. The reference
cavity, sketched in Fig. 4.11, is composed of two mirrors, mounted on a solid spacer. The
latter is made of Zerodur28, which provides a low relative thermal expansion coefficient
of 0 ± 1 × 10−7/K. The reference cavity is placed in a vacuum chamber to prevent
27 Op5 AOM-Driver, H. Lenk, University of Mainz.
28 ZERODUR, Schott AG.
62
4.3. The laser system
fluctuations of its optical length due to air convection and temperature changes. To
further reduce thermally induced fluctuations of the cavity length, the Zerodur-spacer
is placed on two ceramic holders, which provide a decoupling from the environment
with a thermal conductivity of < 1 J/(Kms). The mirrors of the cavity are in a stable
hemispherical configuration: one plane mirror and one concave mirror with a radius
of curvature of 250mm are mounted at a distance of 100mm. Both of them provide
a reflectivity of 99.0(5)% for the relevant wavelengths29. The cavity has a calculated
finesse of F ≈ 350 and a free spectral range of FSR = 1.5GHz. One mirror is mounted
on a stack of two counter-acting piezo-rings, which compensate each other for thermal
fluctuations. A high voltage control30 of the piezo allows for tuning the length of the
cavity in order to match a TEM00 mode to the desired wavelength close to the resonance
frequency of the ion. We use a PDH locking scheme to stabilize the laser frequency to
the resonance line of the cavity. A simple CCD camera31, as well as a photo diode are
available to monitor the transmission signal of the cavity modes.
For the 397 nm laser, the error-signal from PDH lock on the reference cavity is low-pass
filtered and fed to a digital PI-regulator32. The correction signal controls the piezo which
adjusts the angle of the grating of the laser diode. This locking scheme compensates for
slow drifts of< 10 kHz. To level out fast frequency fluctuations, a second PID regulator33
with a bandwidth of 100MHz controls the JFET on the current modulation of the laser
diode. However, the portion of the intensity coupled to the SHG-cavity, and thus its
output power, depends on the relative frequency deviation between the resonance of the
SHG-cavity and the frequency of the laser diode. To reduce these fluctuations, the fast
PID controller of the laser diode uses the error signal from the SHG-cavity, instead of
the one from the external cavity. Slow drifts are compensated by the piezo controlled
length of the SHG-cavity, see Sec. 4.3.2. For the 866 nm laser we use the error-signal of
the reference cavity as input for both, the slow PI regulator, controlling the piezo on the
grating, as well as the fast PID-regulator to level the current in the diode.
29 batch number D214K008, Layertec GmbH.
30 EBS8030, iseg Spezialelektronik GmbH.
31 USB-webcam VC2PMA, Conrad Electronic, lens removed.
32 DigiLock 110, Toptica Photonics AG.
33 PDD 110, Toptica Photonics AG.
63

5 Coherent excitation of motion
The unique properties of the funnel-shaped ion trap are the basis for the operation
of the heat engine experiments. Here, we provide a characterization of the effect of
the tapered trapping potential on the motional behavior of the ion. Furthermore, we
present preparatory measurements which are prerequisites for the characterization of
the single-atom heat engine.
We start this chapter with a presentation of the measurement techniques for a precise
determination of the trap frequencies of the ion in Sec. 5.1. These techniques are used to
characterize the effect of the tapered potential by measuring the radial trap frequency
of the ion as a function of its position along the trap axis. As direct consequence of the
tapered potential, we demonstrate in Sec. 5.2 the fundamental elements where the heat
engine functionality is based on: a change of the extent of the motional state of the ion in
radial direction leads directly to a shift of its equilibrium position along the trap axis. We
will demonstrate this effect using a coherent oscillation in radial direction, in combination
with a precise determination of the ions position from camera images. During the heat
engine operation, the ion is running in a steady state, where a controlled dissipation
balances the engine’s output power. Consequently, the dissipation confines the resulting
oscillation of the ion and is indispensable for a continuous heat engine operation. In
Sec.5.3 we show that laser cooling leads to a controlled dissipation and present a detailed
study of two different approaches to determine the corresponding damping coefficient.
Furthermore, we show in Sec.5.4 that, as a consequence of the tapered potential, each ion
in a linear ion crystal provides an individual radial trap frequency. This feature allows
us to address the vibrational modes of the ions locally.
Parts of this section, comprising the characterization of trap as well as the demonstra-
tion of the working principle of the heat engine, have been published in:
Nano-Motor aus einem einzelnen Atom,
J. Roßnagel, K.N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler and K. Singer,
GIT Labor-Fachzeitschrift 6, 51 (2014).
5.1. Characterization of the ion trap
5.1.1. Measurement of trap frequencies
The fundamental tool for the characterization of the ion trap is the measurement of the
trap frequencies of the ion. The underlying principle is to drive the harmonic oscillator
65
5. Coherent excitation of motion
a) 0b)
8
amplitude
phase shift
6
2 A p
2
4
2
-p
77 78 79 80 81 82 83 84
Driving frequency wd (kHz)
Figure 5.1. – Resonance of the harmonic oscillator. (a) Measured steady state amplitude D of the
oscillation in axial direction z as a function of the driving frequency ωd, measured with
continuous damping by laser cooling with 30 µW. The fit of Eq. (5.1) (blue line) to the data
(blue dots) reveals a resonance frequency of ωz = 80.86(1) kHz, a damping coefficient of
γ = 1950(60) s−1, and an acceleration Fd/m = 17.1(4)× 103 ms−2. The corresponding phase
shift φ (green) between the drive and the oscillation of the ion goes from φ = 0 for ωd ≪ ωz
to φ = −π for ωd ≫ ωz. (b) Camera image of an oscillating ion. The projection of the
fluorescence distribution on the direction of oscillation shows the characteristic behavior from
Eq. 5.4. Each point in a) corresponds to an individual camera image similar to b).
by an oscillating perturbing force with a variable driving frequency ωd, and to analyze
the oscillation amplitude from camera images. For a weakly damped harmonic oscillator,
γ ≪ ωz, the steady state amplitude A of the oscillation is governed by the driving force
Fd and the dissipation γ, and reads
F /m
A(ωd) = √ d , (5.1)
(ω2 −ω2)2i d + (2γωd)2
w{ ith the os}cillating mass m. The amplitude A peaks in a resonance if ωd = ωi, with ωi ∈
ωx,ωy,ωz . Here, we are only interested in the relative amplitude, whose maximum
reveals the resonance frequency ωi, as shown in Fig. 5.1 (a). When scanning over the
resonance, the phase shift between the driving signal and the oscillation of the ion
changes according to ( )
2ω γ
φ(ωd) = arctan
d
− 2 . (5.2)ω2z ωd
Consequently, the resulting time resolved trajectory of the driven damped harmonic
oscillator at steady state reads
xi(t) = A(ωd) · sin (ωdt− φ(ωd)) . (5.3)
We use two different techniques to drive a coherent oscillation of the ion:
i) We superpose the electromagnetic trapping potential with an oscillating electric field
by connecting a signal generator to one of the trap electrodes. The amplitude of the
oscillation is evaluated as a function of the driving frequency ωd by analyzing camera
images. For exposure times longer than the oscillation period, we observe spatially
smeared fluorescence images of an oscillating ion, as shown in Fig. 5.1 (b). The projection
66
Amplitude A (µm)
Phase shift f
5.1. Characterization of the ion trap
Figure 5.2. –Measurement of the eigenfrequencies of an ion by a modulated laser beam. (a) Mod-
ulation signal at 170 kHz on a photo diode. (b) Measurement and fit of the axial trap fre-
quency resulting to ωz = 170.697(2) kHz. (c) Same for the radial eigenfrequencies revealing
ωx = 488.350(5) and ωy = 491.31(1).
of the fluorescence distribution Fm on an axis parallel the direction of the oscillation is
well described by a time-integrated and sinusoidally modulated Gaussian,
∫ [ ]π/2 2
Fm(x) ∝ exp −
(x−D sin φ)
dφ. (5.4)
−π/2 σ2
Here, the modulation depth D corresponds to the steady state amplitude A of the
coherent oscillation. The width of the Gaussian σ is determined from the fluorescence
distribution of an ion at rest. An exemplary measurement of the resonance of a trapped
ion is shown in Fig 5.1. A fit of Eq (5.1) reveals the resonance frequency ωi of the ion.
ii) We excite an oscillation of the ion by a modulation of the intensity of the cooling
laser. This leads to a modulation of the radiation pressure on the ion, and thus to an
oscillating driving force [Dre04]. The intensity modulation is achieved by switching the
AOM-driving on and off at a frequency ωd. When the cooling laser is pulsed, likewise the
recorded intensity of the fluorescence signal is. If the pulses are short as compared to one
oscillation period, the ion is illuminated only at one specific phase of its oscillation, which
is measured and evaluated in Fig. 5.2. For the non-resonant case, the phase φ between
the oscillation and the driving signal is φ = 0◦ or φ = 180◦. The laser pulses reach the ion
at a turning point, but the amplitude of the oscillation vanishes. When approaching the
resonance, the phase change is negligible but the amplitude of the oscillation increases,
the turning points move outwards. When ωd is close to the resonance ω0, φ changes
from 0◦ to 180◦ and consequently the fluorescence signal switches from the one turning
point to the other. The center of gravity of the detected fluorescence distribution z f is
well modeled by the function
αz(ωz (ω ) = d
−ω0)
f d (ωd −ω 20) − 2
+ z0, (5.5)
Γ
67
5. Coherent excitation of motion
480
x-direction
y-direction
460
440
420
400
-200 -100 0 100 200 300 400 500
axial position (µm)
Figure 5.3. –Measured trap frequencies ωr in both radial directions as a function of the relative
position along the z-axis, measured with technique i) (see text). A fit of Eq. (2.17) to the data
reveals the absolute position of the ion inside the trap.
where the constant αz is related to the amplitude of the oscillation and Γ is the width of
the resonance [Dre04]. Consequently, this technique provides a precise measurement of
the resonance frequency of the ion.
There are two essential differences between the techniques i) and ii). One is the
evaluation procedure: in i) broad and smeared out fluorescence distributions are fitted,
resulting in an uncertainty of 102Hz. In ii) we evaluate the shift of the turning points by
fitting a localized fluorescence image, resulting in an improved uncertainty of 101Hz.
The second difference is in the driving operation: ii) is based on a modulation of the
scattering rate of photons and may influence temperature and dissipation. In contrast,
the modulation of the trapping potential in i) does not change the interaction of the ion
with light fields.
5.1.2. Trap frequencies in a tapered potential
We measure the radial trap frequencies ωx,y(z) at different positions z along the trap
axis to characterize the coupling between the radial and the axial modes of oscillation,
Eq. (2.17), experimentally. The results are shown in Fig. 5.3, revealing the strong depen-
dence of ωx,y on z, as well as its 1/z2 behavior. From a fit of Eq. (2.17) to the measured
data we obtain ωx0/2π = 447.4(5) kHz and ωy0/2π = 451.1(5) kHz. The lifted degen-
eracy of the two radial trap frequencies is achieved by an additional static electric field.
This measurement allows to obtain the radial distance r0 of the ion to the trap electrodes
at the position z = z0. Averaged over both radial directions, the fit of Eq. (2.17) yields a
geometry factor of
tan θ rad
= 154(3) =⇒ r0 = 1.19(2)× 10−3m. (5.6)r0 m
By taking the geometry of the trap into account (see Sec. 4.1), r0 reveals the absolute
position of the ion at a distance of z0 = 7.7(8)× 10−4m from the endcap at the narrow
side of the trap. The position of the ion has been chosen on the narrow side, since the
coupling of the modes scales with the distance of the ion to the electrodes, according to
68
radial trap frequency (kHz)
5.2. Demonstration of the elements of the heat engine working principle
Figure 5.4. – Demonstration of the fundamental working mechanism of the heat engine. (a) Pro-
jection of the fluorescence distribution of the ion on the radial direction when driven har-
monically. The increase of the amplitude of the oscillation with a radial trap frequency of
ωr/2π = 468kHz has been measured from 0 to 22 µm by fits of the integrated probability
distribution of coherent states. (b) Projection of the integrated fluorescence distribution on
the axial direction, corresponding to the measurements in a), at an axial trap frequency of
37kHz. (c) Evaluation of the shift of the ion along the trap axis as a function of its extent in
radial direction, using trap frequencies of ωz/2π = 90kHz and ωz/2π = 37kHz. For each
point, the mean phonon number n of the radial coherent excitation is given, as well as and the
axial shift in multiples α of its ground-state wave-packet size. A quadratic fit, resulting from
the 3-dimensional harmonic potential shape, represents well the measured data.
Eq. (2.17). Knowing the precise position of the ion allows us to obtain precise numbers
for the trapping potential and its higher momenta. These numbers are essential for the
analytical calculation of the expected force, power and efficiency of the engine in Ch. 7.
5.2. Demonstration of the elements of the heat engine
working principle
In order to demonstrate the elements of the working principle, we analyze the shift of the
position of the ion along the trap axis as a function of the width of its radial distribution.
The particular kind of the radial state does not mat(ter, since th)e trapping potential
is sensitive only on its variance in radial direction, 〈x2〉+ 〈y2〉 . Using a resonant
modulation of the voltage on the outer electrodes, we induce a coherent oscillation of the
ion in radial direction. Such a coherent oscillation allows to achieve large amplitudes of
69
5. Coherent excitation of motion
some tens of micrometers with a high degree of control. As shown in Sec. 2.5, a change
in the variance of the radial state leads to a shift of the equilibrium position of the ion
along the trap axis.
In Fig. 5.4 we compare the projection of the fluorescence distribution of the coherent
oscillation at different amplitudes with the corresponding static shift along the trap
axis. At trap frequencies of ωz/2π = 37 kHz and ωr/2π = 468 kHz, an oscillation
amplitude of D = 38 µm in radial direction leads to a shift of ∆z = 22 µm. To obtain a
quantitative correlation, we have evaluated those measurements for different axial trap
frequencies, which are shown in Fig. 5.4 (c). The expected quadratic behavior ∆z ∝ D2,
see Sec. 2.5, is well reproduced. These measurements demonstrate the effects of the
coupling of the radial and the axial mode of oscillation as a consequence of the tapered
trapping potential. For the heat engine operation, this mechanism translates a change of
the ions temperature (variance of the radial state) into mechanical work (shift in axial
direction) [Tol14].
5.3. Determination of the damping coefficient
In order to provide damping to a coherent oscillation in the axial degree of freedom,
we apply a cooling laser along the trap axis. The according damping coefficient γ is
controlled by tuning the laser parameters. Knowing γ precisely allows to extract the
driving power from the amplitude of a steady state oscillation. Additionally, a thermal
excitation of the axial mode originates from spontaneous decay through laser cooling.
During the heat engine operation, the incoherent, thermal excitation in axial direction
may conceal the coherent oscillation of the ion, why a cooling process along this direction
becomes necessary.
In order to provide efficient cooling whilst maintaining controlled but low damping of
the oscillation, the axial cooling laser is applied only around the turning points of the
axial motion, see Fig. 5.5 (a). At the turning points, the center of mass velocity of the
ion vanishes and the laser is cooling the thermal excitation only [Met99]. Extending the
cooling period results in an increasing damping of the coherent oscillation. The duty
cycle of the pulsed dissipation laser is chosen to be 20%, if not otherwise noted. This duty
cycle provides a damping which contains the driven coherent oscillation typically within
some 10 µm during heat engine operation. Regarding the thermal excitation, it provides
enough cooling to counteract the heating processes, while the power of the dissipation
laser is not saturating the S1/2 − P1/2 transition. Note, that a saturation of the S1/2 − P1/2
transition by the dissipation laser would change the scattering rate of the cooling laser
for the thermal bath interaction and would influence the resulting temperatures.
In the following we will present two independent ways to determine the damping and
to characterize its behavior.
5.3.1. Step response to coherent drive
The first method employs the step response of the ions motion in the axial direction to a
coherent drive, under the influence of damping. At time t = 0, we start to drive a Doppler
cooled ion in axial direction with a drive frequency ωd close to its resonance frequency ωz.
A rectangular voltage modulation is applied to the end-caps, which is controlled by an rf-
70
5.3. Determination of the damping coefficient
Figure 5.5. – (a) Laser cooling is applied around the turning points (blue) of the oscillation. The duty
cycle of the cooling pulses over one oscillation period is 0.2. (b) Simulated step response of
the ion’s motion to a coherent drive. The gray line shows the time resolved oscillation of the
ion. A detuning of the driving frequency ωd to the resonance frequency ωz of ∆ω = 1.5kHz
leads to a beat signal, reflected by the envelope function (orange), which decays through
dissipation and reaches a steady state with amplitude A. The effect of the damping coefficient
γ is illustrated by three different envelope functions for γ = 700 s−1 (blue), γ = 1500 s−1
(orange) and γ = 3000 s−1 (green).
switch34. For t > 0, the displacement D(t) of the coherent state increases until it reaches
a steady state between drive and the dissipation with an amplitude A. Focusing on the
trajectory of the ion, D(t) represents the envelope function of the resolved oscillation
of the ion. Using the EMCCD camera, the exposure time τexp cannot be reduced below
1ms and is thus much longer that the oscillation period of the ion, τexp ≫ 1/ωz. The
acquisition process thus integrates over many oscillations and, as a consequence, D(t) is
the only accessible observable.
The time-resolved trajectory of the ion under step response reads [Dem08]
[ ]
ω
z z(t) = A0 sin(ωzt)− sin(ωdt) e−γt , (5.7)ωd
and is illustrated in Fig. 5.5 (b). Right after starting the drive, t & 0, the ion starts to
oscillate with its resonance frequency ωz. To resolve the the frequency of the driving
signal, a time t ≫ ω−1d is needed. The interference betweenωd andωz leads to a beat note
at ωb = ωz −ωd. Due to the dissipation, the initial oscillation with ωz decays in company
of the beat, and for t ≫ ω−1d the ion oscillates with ωd. It reaches a steady state amplitude
A(ωd,γ, Fd) according to Eq. (5.1), which is governed by the driving force Fd and the
damping coefficient γ. The envelope function D(t) of the ion’s oscillation is obtained by
a frequency decomposition of the beat signal into a fast and a slow oscillation [Dem08],
ωz +ωc = d and ωb = ωz −ωd.2
The frequencies ωz and ωd can thus be rewritten as
ω
ω b
ω
z = c+ and ωd = c− b .2 2
34 ZYSWA-2-50DR, Mini-Circuits.
71
5. Coherent excitation of motion
a) 8 b)
6
4
2 D
-20 -10 0 10 20
Position z (µm)
0
0 2 4 6 8 10 -10 5 0 5 10 15
Time t (ms) Time (µs)
Figure 5.6. – (a) Determination of the damping coefficient γ. The step response is evaluated by
determining the amplitude A of the ion’s oscillation (orange circles) at time t after switching
on the end-cap drive. A fit to the data (orange line) reveals γi = 520(100) s−1 and a beat-note
of ωb/2π = 190(10)Hz. Inset: Determination of oscillation amplitude at 5.5ms by fitting
Eq. (5.4) (black line) to the projection of the corresponding camera images (gray circles). (b)
Timing of the cooling pulses. The rectangular driving signal (blue), measured just at the
vacuum feed-throughs, is deformed through the high pass filter properties of the transformer,
cf. Fig. 4.5. To deduce the phase of the ion, this signal is fitted by a cosine (orange). On
resonance, the harmonic oscillation of the ion follows with δφ ≈ −90◦ phase shift (violet). The
photo-diode signal of the dissipation laser (green) is tuned to the turning points by a delay of
2.1 µs with respect to the driving signal.
Using Euler’s formula eiφ = cos φ+ i sin φ → sin φ = Im[eiφ], we can rewrite Eq. 5.7 as
[ ] [ ]
z(t) = A Im ei(c+ωb/2)t − ωz e−γtei(c−ωb/2)t0 = A0 Im eicteiωbt −
ωz e−γteict ,
ωd ωd
which is in trigonometric representation
[ ( ) ]
ω
z(t) = A0 sin(ct) cos(ωbt − z) e−γt + cos(ct) sin(ωω bt) .d
However, we are interested in an analytical description of the time depending amplitude
D(t) of the oscillation, not in the resolved motion of the ion z(t). In phase-space, the
fast oscillating components sin(ct) and cos(ct) are perpendicular. The amplitude D(t) is
then defined by the length of the phase-space vector, which is calculated according to the
Pythagorean formula
√( )
D ω
2
(t) = A cos t z(ω )− e−γtb + sin(ω t)2b . (5.8)ωd
The envelope function D(t) is plotted in Fig. 5.5 (b) for different values for γ and is
compared to the time resolved oscillation z(t).
In order to determine γ in the experiment, the amplitude D(t) can be extracted from
the fluorescence distribution on camera images at different times t after switching on the
coherent drive. We employ an exposure time τexp = 1ms, which is much longer than the
oscillation period of the ion, τosc = 2π/ωz = 12.3 µs, but still short enough to resolve the
dynamics of the step response. An example of a step response measurement is shown in
Fig. 5.6 (a). Each point is determined by fitting Fm, Eq. (5.4), to the projected intensity
72
Displacement D (µm)
Intensity (a.u.)
Voltage signal (a.u.)
5.3. Determination of the damping coefficient
Figure 5.7. – Determination of the damping coefficient γ by a measurement of the displacementD(t)
as a function of time. (a) D(t) for different power Pd of the pulsed dissipation laser at a duty
cycle of dd = 0.2: Pd = 17mW (blue), 46mW (orange), and 60mW (green) (power measured at
dd = 1). The detuning of the driving signal has been set to ωb/2π = 0.25 kHz. The uncertainty
of D is of statistical nature, while the error bars along the time axis represent the exposure
time of 1ms. A fit to the data reveals γ = 220(60) s−1, 650(100) s−1, and 1000(250) s−1,
respectively, where the uncertainty of γ stems from the fitting error. (b) Resulting γ-values
from a), revealing a linear trend γ(Pd) = 15(1) · Pd (s · µW)−1. (c) D(t) for different ωb at
constant γ. The resonance frequency is measured to be ωz = 81.5kHz and the resulting
damping coefficient stays in the range γ = 700 ± 100 s−1. The driving power has been
adjusted to achieve approximately the same steady-state amplitude for all ωb. For the sake of
clarity we don’t display the time error-bars.
distribution, as shown in the inset of Fig. 5.6 (a). The function D(t), Eq. (5.8), is fitted to
the measured amplitudes and reveals the damping coefficient of the axial dissipation
laser.
To calculate the right timing of the laser pulses, we approximate the effective driving
signal by a cosine function, see Fig. 5.6 (b). The ion follows the corresponding oscillation
of the electric field with a π/2 phase shift [Dem08]. Additionally, the charging process
of the capacitive load of the bias tees and the electrodes leads to a lag of the electric field
oscillation inside the trap behind the driving signal, which is calculated in Sec. 7.3 and
results to 800 ns.
In general, the strength of the dissipation can be adjusted by varying the detuning ∆νd
and the power Pd of the axial laser, by changing the timing of its pulses with respect to the
ion’s oscillation, or by modifying its duty cycle. For practical reasons, we use the power
of the axial dissipation laser Pd to tune γ, as shown in Fig. 5.7 (b). We observe a decrease
73
5. Coherent excitation of motion
Figure 5.8. – Broadening of the spectral resonances of the oscillation of the ion for different damping
coefficients. The steady-state amplitude A is recorded as a function of the driving frequency
ωd for different strengths of the pulsed dissipation laser. The fit of Eq. (5.1) (solid line) to the
data (dots) reveals damping coefficients of γ = 550(40) s−1 and γ = 180(20) s−1.
of the decay-time of the beat signal when increasing Pd from 17mW to 60mW, where Pd
is measured on a continuous beam instead of the pulsed beam35. The corresponding γ is
thereby tuned from 200 to 1000 s−1.
To characterize the influence of the beat signal, we have varied the driving signal ωd
stepwise between the measurements, see Fig. 5.7 (c). The best precision for an analysis
of γ is achieved for ωb being small but non-vanishing. Then, not only the rise time of
D(t) but also the decay of the beat signal can be included to the fitting procedure for γ.
However, if ωb ≫ γ the laser pulses are not applied anymore in the turning points of the
oscillation, because of the additional phase shift for a non-resonant drive, see Eq. (5.2).
The effective detuning of the dissipation laser is then changed, since it interacts with
the ion at higher velocities. This effect will be discussed in more detail in the following
section.
Note, that the inverse decay of the oscillation can be used equally to evaluate γ, where
the decaying amplitude is measured after stopping the drive. In this case we measure an
exponential decay of D(t), again superposed with a beat between the driving frequency
and the eigenfrequency of the harmonic oscillator. However, since the beat is then on
low amplitudes, in contrast to step response presented before, the contributions of the
decay and of the beat are hard to differentiate. Although the resulting values of γ are
comparable, the precision of the results is strongly reduced.
5.3.2. Broadening of the resonance
A complementary method to determine γ is to observe the steady-state oscillation
amplitude as a function of the driving frequency. As shown by Eq. (5.1), the width as
well as the height of the spectral resonance profile is governed by γ. The amplitude of the
oscillation is again determined from projections of the camera images. A measurement
of the steady-state amplitude A(ωd) is shown in Fig. 5.8. Compared to the step-response,
this technique is more robust against fluctuations and is less demanding on the camera
image acquisition. It allows longer exposure times and thus enables us to characterize
35 The reduced integrated power of the pulsed beam easily falls below the background threshold.
74
5.3. Determination of the damping coefficient
Figure 5.9. – Characterization of the damping coefficient γ as a function of the parameters of the
dissipation laser. (a) Variation of the detuning ∆ν from the atomic resonance at a laser power
of Pd = 23 µW. Eq. (5.9) has been fitted to the data, and we obtain a maximum damping
coefficient at resonance γ(∆ν = 0) = 390(80) s−1. (b) Variation of the laser power Pd, at a
detuning ∆ν = −9(3)MHz. The fit of Eq. (5.10) approximates well the measured data and
reveals a saturation power of P0 = 27(9) µW. Both data sets have been measured by the
evaluation of the broadening of the resonances.
the dependencies of γ on external parameters with a reasonable investment of time.
However, when scanning the driving frequency while using a pulsed dissipation laser,
the phase shift φ(ωd), Eq. (5.2), becomes important. It leads to a deviation of the cooling
pulses from the turning points of the ion’s trajectory. As a consequence, the ion has a
higher velocity when interacting with the laser beam and thus the effective γ is changed.
For small γ and thus steep phase changes, we observe a reduced amplitude on the flanks
of the resonance signal, see Fig. 5.8, which may stem from the phase mismatch. However,
it seems that these deviations are small and Eq. (5.1) is still a good approximation for the
measured resonances.
We investigate the spectral broadening of the resonances to characterize γ in more
detail. We therefore record the resonances as a function of the detuning ∆νd and the
power Pd of the dissipation laser.
In a simplified model, we expect γ to be proportional to the scattering rate of photons,
γ ∝ ssc(∆ν), since laser cooling is the origin of dissipation. ssc(∆ν) corresponds to the
spectral lineshape of the S1/2 − P1/2 resonance [Ste86a],
(Γ /2)2
ssc(∆ν,t) = s
P
0ν 2 2 . (5.9)(∆ν+ ∆νD(t)) + (ΓP/2)
with the natural linewidth ΓP = 23.1MHz, see Sec. 2.7. Due to the oscillation of the ion
the line is Doppler shifted by ∆νD(t) = ∆νD0 sin(ωzt). The maximum Doppler-shift for
an amplitude of 10 µm results to ∆νD0 = ±13MHz, at a trap frequency of ωz = 81 kHz.
However, the damping laser is applied around the turning points and within this period
the according Doppler shift rages between -4 and 4MHz. It cancels, if we approximate
the resonance line to be linear on this scale, since the velocity is symmetric around the
turning points. Fig. 5.9 (a) shows the measured data γ(∆ν) with a fit of Eq. (5.9), which
reproduces well the general behavior of γ(∆ν). Here, ∆ν has been measured by the
wavemeter, which is subject to strong fluctuations on the MHz scale. As a consequence,
and because of the scattering of the measured γ values, Fig. 5.9 (a) gives a rough idea of
the influence of ∆ν on γ, but does not allow to confirm or disprove the validity of our
simplified model.
75
5. Coherent excitation of motion
Regarding the dependance on the laser power Pd, we expect by ourmodel an increasing
γ for increasing Pd. However, for high laser power, the S1/2 − P1/2 transition is saturated
and γ may reach a plateau. The analytical description therefore follows the exponential
approach ( )
− Pd
ssc(P Pd) = s0p 1− e 0 . (5.10)
Fig. 5.9 (b) shows ssc(Pd) fitted to the measured data γ(Pd). Again, Eq. (5.10) reproduces
well the general behavior of γ(Pd), without allowing for substantiated statements about
the model.
This qualitative analysis of γ gives a good idea of its behavior, but it can’t be com-
plemented by quantitative statements, since γ(∆ν, Pd) is a function of both, power and
detuning of the laser. For a complete analysis a 2-dimensional scan of γ in the (∆ν, Pd)
plane should be recorded, which would exceed the scope of our study. Nontheless, the
presented techniques are adequate to determine γ for the specific situation during the
heat engine experiments, which are presented in Ch. 7.
5.4. Addressing of single atoms and controlled interaction
Finally, we present another feature of the tapered trapping potential: due to the asym-
metry in the trapping potential each ion in a linear chain of ions provides an individual
radial trap frequency. Consequently, the oscillation of each ion can be addressed sepa-
rately by an external drive. If the driving frequency is resonant to one of the ions, the
other ions still couple through Coulomb-interaction to the oscillating ion, but do not
experience a resonant excitation. Fig. 5.10 shows this effect qualitatively in a linear 7
ion crystal, where the oscillation amplitude decays with 1/d2 with the distance d to the
driven ion. If the inter-ion distance is large and the spectral dissipation-broadening of
the resonances is reduced to a minimum, the excitation of common modes is suppressed
completely and only the ion which is resonant to the external drive oscillates. The ad-
dressing of single ions, as well as the decoupling of the common modes of oscillation,
has been demonstrated in a three and in a four ion crystal in Figs. 5.11 (a) and (b), where
the axial trap frequency has been set to 40 kHz and 30 kHz, respectively.
The fact that in a tapered trapping potential the coupling between the modes of
neighboring ions can be tuned deliberately through the ion distance and their dissipation
promises advantages when studying, for instance, the transport of heat or energy along
ion chains [Lin11, Ber13, Ram14]. A study of such phenomena has failed so far because
of technical difficulties and fast thermalization of the modes of oscillation. A strongly
reduced interaction facilitates such investigations. Furthermore, the addressing of single
ions may provide strong advantages for quantum computation with ion strings, as we
discuss in Sec. 8.4.
76
5.4. Addressing of single atoms and controlled interaction
Figure 5.10. – Camera image of a linear ion crystal of 7 ions. The excitation frequency is set to
the resonance frequency of the left ion. The Coulomb-interaction, together with a high
dissipation broadening of the resonances, leads to an oscillation of the neighboring ions, but
with an amplitude proportional to the interaction range 1/d2, with d being their distance
from the oscillating ion.
Figure 5.11. – (a) Individual addressing in a linear chain of 3 ions. The driving frequency has been
scanned over a range of 20 kHz, addressing the resonance of only one ion at once. Each of
the curves corresponds to the integrated fluorescence signal of one of the regions of interest
(ROI), indicated on the camera images. The fluorescence signal for an oscillating ion is
slightly reduced due to the additional Doppler shift. The difference in the distance of the
individual resonance frequencies is a result of the 1/z2 dependance of the trap frequency
as discussed in Sec. 5.1.1. (b) The same measurement with a linear crystal of 4 ions. The
contrast of the signal is reduced, but the ions are still addressed individually.
77

6 Incoherent thermal excitation
This chapter is dedicated to the questions: "What is temperature, how can we control it,
and how can we measure it?" The notion of temperature is clear for macroscopic systems,
but becomes more complex on the nanoscopic scale. Here, we first want to focus in
Sec. 6.1 on the identification and characterization of thermal states, before we discuss
how to control and tune the temperature of single trapped ions deliberately. Control-
ling and measuring the temperature of single particles is essential for many different
applications. Especially for the measurements of the single-ion heat engine in the next
chapter a fast, precise and non-invasive thermometry technique is mandatory in order
to analyze thermodynamic cycles. Furthermore, there is growing interest in the field
of quantum thermodynamics and techniques like reservoir engineering or dissipative
state preparation have moved into focus of ion trap experiments [Bér12, Toy13, Kie15].
In this chapter we experimentally demonstrate a method for a rapid determination of
the temperature of trapped ions which is suitable for monitoring fast thermalization
processes. The method is based on the multi-level electronic structure and the occurrence
of dark resonances in the fluorescence signal of trapped ions. Narrow dark resonances
originate from coherent population trapping in a three level system [Alz76, Ari76, Ari96]
and provide a high velocity sensitivity of their fluorescence signal.
This new method complements the thermometry toolbox for trapped ions at an
intermediate range between 10−1 and 10+2mK, the typical temperatures of Doppler
cooled and moderately heated ions. For highly excited thermal states there are already
well established techniques, such as Doppler recooling or spectral thermometry. The
Doppler recooling technique is appropriate for a temperatures range 10−1 − 10+1meV
(103 − 105mK) [Eps07, Wes07]. For lower temperatures spectral thermometry and spatial
thermometry have proved to be useful techniques. Spectral thermometry uses the line
shape of the fluorescence signal to extract the temperature of the ion [Nag83], but is
accompanied by rather large uncertainties [Has98, Her09]. Extracting the temperature
from the spatial distribution of a thermal atom is limited by the resolution of the imaging
system [Nor11]. By a precise calibration of the imaging optics and its magnification,
temperatures down to the Doppler limit can be measured [Knü12].
In this chapter we demonstrate that spatial thermometry supplements the dark reso-
nance technique perfectly for higher temperatures. For experiments in the strong-binding
regime ω ≫ Γ (trap frequency ω much larger than transition linewidth Γ) and close to
the ground state of motion, precise temperature measurements can be achieved by spec-
troscopy of the sidebands of an optical transition ν, which originate from the oscillation
79
6. Incoherent thermal excitation
of the ion. The mean phonon number of the vibrational state of the ion is then deduced
from the relation of the excitation probabilities on the red and blue sidebands ν−ω and
ν+ω, respectively [Tur00a, Eps07, Bro15]. Dark resonance thermometry, however, can
be applied in the weak-binding regime ω ≪ Γ, where sidebands are not resolved due to
low trap frequencies, and for thermal states between a few and up to some hundreds of
phonons.
We start the discussion of dark resonance thermometry with a derivation of the ap-
pearance of dark resonances in a multi-level system from the steady-state solution of
the optical Bloch equations in Sec. 6.2. This analytical model is used to obtain tem-
perature values from measured resonances. We show in a semi-classical picture how
dark resonances can be used to tune the temperature of the ions. The laser linewidth
is a crucial parameter for dark resonance thermometry. For both lasers which are used
for spectroscopy, the linewidth is on the same order of magnitude as the broadened
resonances are. As a consequence, any uncertainty of the laser linewidth affects the
precision of the resulting temperatures. In Sec. 6.3 we deduce the laser linewidth from
the error-signal of the PDH locking scheme. While we show in Sec. 6.4 the results of
steady state temperature measurements, Sec. 6.5 focuses on measurements of the dy-
namics of heating and cooling processes, and demonstrates the limits and effects of the
temporal resolution of the measurement. Furthermore, in Sec 6.6, we show how the
ion temperature can be controlled between 0.7mK and more than 10mK by tuning the
cooling laser to selected features of the dark resonances. Finally, in Sec. 6.7 we compare
our method to the spatial thermometry technique. Sec. 6.8 summarizes the results of the
dark resonance thermometry technique and proposes possible applications.
The key features and basic demonstrations of the dark resonance thermometry tech-
nique have been published in:
Fast thermometry for trapped ions using dark resonances,
J. Roßnagel, K.N. Tolazzi, F. Schmidt-Kaler and K. Singer,
New J. Phys. 17, 045004 (2015).
6.1. Temperature of a single trapped ion
6.1.1. The notion of temperature
For a classical, macroscopic and ideal gas system in the thermodynamic limit (large
number of particles), the individual particles are interacting and exchange their kinetic
energy. Each of the particles is experiencing a Brownian motion under random kicks from
the other particles [Uhl30]. Since we observe the system as a whole, not the individual
trajectories but the mean values of the observables are of interest. Through the interaction
of the particles, the distribution of kinetic energy tends towards an equilibrium. The
temperature T is then defined as the mean kinetic energy 〈Ekin〉 in each degree of freedom
f ,
〈 fEkin〉 = kBT. (6.1)2
When heat is added to the system, it is distributed over all particles and all degrees of
freedom and the system reaches again a thermal equilibrium.
On the other hand, for quantum systems there is clear definition of the temperature as
well. In Sec. 3.1.1 we have presented the phonon distribution for a quantum harmonic
80
6.1. Temperature of a single trapped ion
oscillator in thermal equilibrium. We have shown that its temperature can be derived
from the mean value 〈n〉 of many measurements of its phonon number according to
T(〈 〉 (h̄ωn ,ω) = 〈 〉 ) (6.2)kB ln n +1〈n〉
Now we want to compare this to microscopic and nanoscopic systems in the classical
weak binding regime, kBT ≪ h̄ω, and thus to the situation in our experiment. In
those systems, a classical trajectory of single particle is observed as a function of time.
However, the temperature is a statistical quantity and thus it is impossible to assign a
temperature to a single particle for a particular moment. The ergodic hypothesis states
that for thermodynamic systems the ensemble average of a large system equals the time
average of a single particle. Thereby, the statistical definition of temperature is translated
to single-atom systems. In contact with a thermal bath, the single particle experiences
fluctuating disturbances of its trajectory and therefore exhibits Brownian motion. As a
consequence of the equipartition theorem, the temperature T of a single atom can be
defined as the mean energy of this thermal Brownian motion, and therefore corresponds
to the mean squared displacement σ
2 kBT 2 1
∫ τ
σ = t with σ = R(t)2 dt, (6.3)
ρ τ 0
where R is the displacement of the Brownian trajectory from its starting point, ρ is a
constant and proportional to the friction and τ is the observation time [Uhl30, Mar15].
This definition of temperature becomes intuitively clear for the situation of colloidal
microparticles immersed in water, as used in many thermodynamic experiments [Bli12,
Bér12, Mar16]. These particles are in permanent contact with the surrounding water
molecules. Observing the microparticle therefore equals the observation of one single
particle of a large ensemble in thermal equilibrium. Considering the equipartition
theorem and the ergodic hypotheses, there is no doubt that the Brownian motion of this
particle reflects the temperature of the system.
For isolated systems, such as a single trapped ion or a levitated microparticle in
vacuum [Mil14], the situation is more complex. In such systems, there is no physical
contact with a thermal heat bath. To define and control the temperature for such systems,
we have to mimic the bath interaction by external means. How would the particle
behave, if it would be in contact with a thermal heat bath? It would exhibit a Brownian
motion, as the microparticle in water does. In our case, the ion is confined in a harmonic
potential, which leads to a normal distribution of the time averaged displacement and
velocity of the ion [Bla86, Mee96]. Consequently, σ is the width of the thermal probability
distribution p of the ions displacement x,
√ [
1 − ](x− x 2
p(x) = exp 0
)
, (6.4)
2πσ2 2σ2
and the temperature of the ion in the weak reads
mω2σ2
T = . (6.5)
kB
Fig. 6.1 shows the measured probability distribution of the recorded fluorescence of a
trapped ion at different temperatures. A fit of p(x) ∝ exp [−x2/σ2] to the data and the
81
6. Incoherent thermal excitation
Figure 6.1. – Spatial intensity distribution of the fluorescence signal of the ion, while in contact
with thermal baths at different temperatures, together with fitted Gaussian distributions.
The residuals demonstrate the agreement of the data with the fitted function. We observe
little deviations at the lower edges which are seen on all ion images and result from halo-like
aberrations of the camera objective. These effects don’t scale with the extent of the fluorescence
distribution and thus disappear for larger distributions.
corresponding residuals demonstrate the Gaussianity of the fluorescence distribution.
However, the Gaussian distribution is a necessary, but no sufficient condition for proofing
a thermal state. There might be other states, e.g. squeezed states, whose time-averaged
projection has a Gaussian shape as well. But, at least, a Gaussian distribution excludes a
variety of different motional states, most important coherent states which are likely to be
excited. To guarantee a purely thermal state of the ion we need to know the nature of
the applied excitation. If this excitation is completely incoherent, provides a Gaussian
probability distribution and is irreversible, a non-thermal state of the ion can be excluded.
Consequently, such an environment mimics an effective thermal heat bath for the trapped
ion [Cir94b, Poy96].
6.1.2. Generation of thermal states at various temperatures
We initialize our experiment by laser cooling the ion. The interaction of a harmoni-
cally bound two-level atom with a laser field is well-understood [Cir92]. In a classical
description, a laser-cooled atom experiences random recoils from photon absorption
and emission events [Ste86a]. The isotropic emission of photons leads to a random
Brownian motion. By the nature of the incoherent spontaneous emission process, all
prerequisites for the generation of a thermal state are met. In a more sophisticated
analysis, the master equation describing the dynamics of a harmonically bound two-level
atom with a laser field is identical to that of a damped harmonic oscillator coupled to a
thermal bath [Cir94b, Poy96]. The corresponding stationary state is hence thermal. In the
following experiments we will employ laser cooling for generating cold thermal states or
to decrease the temperature of hot thermal states in a controlled way.
82
6.1. Temperature of a single trapped ion
Figure 6.2. – (a) Noise signal as a function of time, with N = 0.5Vrms and 2000 samples. (b)
Histogram of the noise signal including 13 k samples. A fit of a normal probability distribution
shows the Gaussianity of the noise. (c) Auto-correlation-function of the noise signal. The finite
correlation is due to limited samples size. The auto-correlation demonstrates the high quality
of the quasi-random distribution of the digital signal. (d) Frequency spectrum of the white
noise, showing an equal probability distribution in a frequency range of up to 10MHz.
Unfortunately, the interaction with a laser beam alone is unsuitable for increasing the
thermal energy. For a blue detuned laser beam, the absorption provides an energy gain
as compared to the emission process and dominates the dynamics by increasing the
kinetic energy of the ion. However, the absorption process is velocity dependent and
will amplify any coherent oscillation of the ion parametrically. This results in regular
oscillation of the ion rather than a thermal excitation.
In order to generate thermal states at higher temperatures in a defined way, we
expose the ion to white, Gaussian electric field noise, as it has been shown in other
experiments [Tur00a, Mar15]. In the classical picture, the kicks due to the Gaussian
noise result in a Brownian motion and thus leave the ion in a thermal state at higher
temperature. In particular, the Maxwellian velocity distribution of the ion remains
Maxwellian during the noise interaction. The width of the velocity distribution increases
during the heating process, but the distribution remains thermal. The master equation
for an ion subjected to Gaussian noise is also of the form of that of a harmonic oscillator
coupled to a thermal bath [Fey63]. The linearity of these two master equations ensures
that the Gaussian shape of the phase space distribution is preserved.
However, in principle one could record the noise pattern of the generator and, after
some time, run the same pattern backwards. If we have started with an ion at rest, we
should end up with an ion at rest after the interaction with the noise signal and with
its inverse signal, since a trapped ion is a perfectly isolated system. Additionally, the
electric field noise leads to an ever increasing amplitude. To counteract these problems
83
6. Incoherent thermal excitation
we combine the noise interaction with a simultaneous laser cooling. The velocity de-
pendent cooling rate limits the increasing temperature and the system tends towards
an equilibrium between heating and cooling rate. Furthermore, the additional sponta-
neous emission events scramble the defined Brownian motion of the noise generator,
and an inversion of the trajectory is thus impossible. As a consequence, due to the white
noise spectrum and the Gaussian probability distribution of the electric noise, and in
combination with laser cooling, it acts as a thermal heat bath on the trapped ion.
To guarantee a heating in radial direction only, without affecting the axial degree of
freedom, we need to shuffle the position of the potential minimum randomly around in
the radial plane, without changing the strength of the trapping potential. As discussed
in Sec. 4.1.3 this is achieved by applying the same noise signal with inverted sign
on two opposing outer-electrodes. As a noise source we use an arbitrary function
generator36 which provides noise signals of up to N = 10Vrms. Fig. 6.2 (a) and (b)
show a recorded noise signal together with its histogram. A fit to the data reveals a
Gaussian-like distribution. A slight drop for small amplitudes may result from the
limited resolution during the record of the signal. Since the noise generator is running
digitally, the output signal is a pseudo-random-noise with a bandwidth of 20MHz. The
corresponding normalized auto-correlation function
1 ∫ T
RN (τ) = N (t+ τ)N (t) dt (6.6)T 0
is presented in Fig. 6.2 (c). The resulting delta-function like auto-correlation RN (τ) =
δ(τ) demonstrates that the generator is not repeating a fixed noise pattern but continu-
ously generates random output voltages. This is important when applying electric noise
to a trapped ion, since any signal repetition with a rate close to one of the trap frequencies
would excite coherent oscillations of the ion. The residual correlation of |RN | ≤ 0.02 is
due to the finite size of the recorded sample. Fig. 6.2 (d) shows the frequency spectrum
of the noise. It provides an equal distribution in range of up to 10MHz and thus can be
regarded as white.
We have now shown that laser cooling, as well as the electric field noise, are engineered
thermal environments for the trapped ion. During interaction, the temperature of the
ion is changed and approaches a steady-state temperature, while its motional state stays
thermal. We show in Sec. 6.5 that, indeed, the dynamics of the temperature changes
behave like thermalization processes. As a consequence, these environments mimic
thermal heat baths where the ion is coupled to, while the steady-state temperatures
correspond to the effective bath temperatures [Cir94b, Poy96].
In the experiment we apply the noise in such a way that the trapping potential is
not squeezed but shifted according to the noise signal. As described in Sec. 4.1.3, the
noise signal is fed with opposing signs to two opposing electrodes, and thus shuffles the
potential around, while the trap frequency stays constant. The force F,
~F(r,t) = m~a(r,t) = e ~E(r,t), (6.7)
corresponding to the electric field
~E(r,t) = ∇~ Φ(r) · N (t), (6.8)
36 50MHz Arbitrary Waveform/Function Generator 3390, Keithley Instruments GmbH.
84
6.2. Principles of dark resonance thermometry
exerts kicks on the ion and leads to the Brownian motion. Here, we have decomposed the
potential of the trapping field ~E into a static part Φ and the superimposed noise signal
N (t). The thermal energy corresponds to the time-average of the kinetic energy
1 〈 〉2 1Eth = 〈Ekin(t)〉 = m v (t) = kBT. (6.9)2 2
〈 〉 〈 〉
In the harmonic potential of the trap we find v2(t) = ω a2(t) , which relates T to
N (t) according to
mω 〈 〉 eω 〈 〉2 2 eω 〈 〉T = a (t) = E (t) = (∇~ Φ(x))2 N 2(t) . (6.10)
kB kB kB
Thus the temperature of the ion scales quadratically with the root-mean-square amplitude
of the noise signal Nrms(t), which we refer to as N in the following,
T ∝ N 2. (6.11)
6.2. Principles of dark resonance thermometry
So far dark resonances have been shown to provide a useful tool for compensating
micromotion in ion traps or tuning the cooling rate for single ions [Rei02b, Esc03, Lis05].
Recently they have been used to determine the temperature of laser cooled neutral atom
clouds in the sub-millikelvin regime [Pet12]. We have adopted this technique for the
first time for trapped ion systems. Observing and analyzing the lineshape of those dark
resonances provides a temperature measurement which is accurate over a large dynamic
range and which is well-suited for applications in systems with low trap frequencies. It
can be applied to large ion crystals and to observe fast thermal processes, which typically
occur in the range of 10−4 s. While dark resonance thermometry can be employed within
a wide range of temperatures it is convenient to use and does not require any additional
lasers or technical components as compared to Doppler laser-cooling.
We first need to derive a theoretical model for the line shape of the dark resonances
depending on the temperature in order to analyze the measured spectra. As described in
Sec. 2.7, in our case the spectrum of the 40Ca+-ion can be approximated by a three-level
system with fine structure levels 2S1/2 = |1〉, 2D3/2 = |3〉 and 2P1/2 = |2〉, while the
magnetic field lifts the degeneracy of the electronic states leading to an 8-level system
[Sch95, Obe99]. However, to qualitatively describe coherent population trapping (CPT)
and the appearance of dark resonances we start with the idealized case of a general
three-level system. Later, this model is expanded to an 8-level system with a quantitative
analysis corresponding the experimental parameters.
CPT first was observed as a drop of the fluorescence signal from Na atoms in a vapor
cell [Alz76]. It has been described as a pumping of atoms into a dark non-absorbing
state, created by destructive interference of excitation pathways from two stable levels
to a common excited level [Kha15]. The effect of CPT should not be confused with the
occurrence of electromagnetically induced transparency (EIT), where a strong light-field
induces dressed states of the excited level due to the AC-Stark effect [Fle05]. Both effects
have similar physical principles, based on interferences in a three-level-system, but
differ in many essential points: while CPT uses the interference of both optically driven
transitions to generate a dark state, for EIT only one laser is needed. The interference
85
6. Incoherent thermal excitation
Figure 6.3. – Emergence of dark resonances due to coherent population trapping. (a) Three-level
system with atomic states |1〉, |2〉 and |3〉, and two near-resonant light fields with frequencies
ωa and ωb in Λ-configuration. For ∆a 6= ∆b the fluorescence from |2〉 are observed. (b) If
∆a = ∆b the population is trapped in a coherent superposition of |1〉 and |3〉. This state is
not coupled to |2〉 and consequently no fluorescence is observed. (c) To obtain a spectrum of
resonance fluorescence the detuning ∆a is scanned. If both detunings are equal ∆a = ∆b the
fluorescence shows a drop, the dark resonance.
arises between the doublet of the dressed states. A second probe laser is used to probe
this effect, not to generate it [Kha15].
We now consider three states |1〉, |2〉, |3〉 in Λ-configuration, where two near-resonant
light fields with frequencies ωa and ωb drive the atomic transitions ω12 and ω23, respec-
tively (see Fig. 6.3 (a)). As long as the detunings ∆i of both light fields are different
∆a 6= ∆b the excited state |2〉 is populated, and fluorescence from spontaneous decay is
observed. In the case that ∆a = ∆b a coherent superposition of the states |1〉 and |3〉 is
generated. If the radiation lifetime of |1〉 and |3〉 is much longer than the lifetime of |2〉, a
significant part of the population is transferred into this superposition state and coherent
population trapping is achieved. Since this state is decoupled from the individual driving
fields, no spontaneous decay from |2〉 is observed, resulting in a "dark" resonance [CT11].
When scanning the frequency of one of the two fields while keeping the other constant, a
drop in the fluorescence spectrum of |2〉 → |1〉 is observed (see Fig. 6.3 (b)). Note that the
steady state of the internal dynamics is generally reached much faster than any relevant
external timescales, in particular thermalization processes [Tol14].
For an ion in motion, this dark resonance line is shifted due to the Doppler effect.
Because a thermal state of an ion consists of many different velocity classes, it leads
to a broadening of the dark resonance. Additional broadening is caused by the phase
fluctuations of the laser fields, denoted by a spectral linewidth Γ. However, if the spectral
linewidth of the lasers is known, the measured broadening of the dark resonance feature
allows to determine the temperature of the ion by comparing with calculated spectra.
6.2.1. Calculation of dark resonance spectra
For a quantitative determination of the ion temperature from the measured fluorescence
spectra, a model for the theoretical lineshape is required. Therefore we describe the
dynamics of a general N-level system by a Lindblad master equation, the time evolution
of the density matrix. We consider the Hamiltonian of the system
H = Hdip +Hion (6.12)
86
6.2. Principles of dark resonance thermometry
which is composed of Hdip, the ion-light dipole interaction and Hion, describing internal
states of the ion including the detuning of the two light fields ∆a and ∆b [Fle05, CT11].
The motional state of the ion, not included in the Hamiltonian, will be introduced later
as a spectral Doppler shift of the detunings ∆a,b.
The Lindblad master equation for the density matrix ρ̂ reads [Cir92]
dρ̂ − i= [H,ρ̂] + L(ρ̂). (6.13)
dt h̄
The spontaneous decay of the population of the excited levels is introduced by the
dissipative Lindblad operator
[ ( )]
L(ρ̂) = ∑ Ci ρ̂C†
1
i − C†i Ci ρ̂+ ρ̂C†2 i Ci . (6.14)i
√
where the transition operators Ci = Γpq|q〉〈p| describe the dissipation along the dipole
transitions p → q. They depend on the decay rates according to Fermi’s golden rule
[CT77, Scu97]
8π2 ∣∣ ∣2
Γpq = ∣〈p|D~ |q〉∣∣ (6.15)3ǫ0h̄λpq
with the atomic dipole operator D~ and the transition wavelength λpq.
The spectral widths of the light fields Γa and Γb are included as an additional broad-
ening of the corresponding atomic level |q〉, see [Obe99] and [Tol14]. Phase fluctuations
between the atomic state and the light√field lead to a decay of the dark state and are
modeled by transition operators Ca,b = Γa,b |q〉 〈q| [CT77, CT11].
To obtain the resulting spectrum of the fluorescence light, the steady state solution
dρ̂/dt = 0 is required. In the following we will solve this equation numerically. We
transform Eq. (6.13) into a matrix equation, in which the density matrix ρ̂with dimension
N×N is represented by a vector with N2 entries, where N is the number of atomic levels
involved. Consequently, the master equation with the Lindblad operator Lik in matrix
form is described by
N2dρ { }~ i = ∑ Lik~ρk = 0 with i ∈ 1,...,N2 . (6.16)dt k=1
One of these equations (for simplicity i = 1) is replaced by the normalization condition
Tr(ρ̂) = 1. Finally, to obtain the steady state solution ρ̂0 we solve the homogeneous
equation (6.16) with d~ρ/dt = 0 by numerical matrix inversion. The fluorescence rate F
is then proportional to the population in the excited state(s) j
F ∝ ρ̂0jj. (6.17)
To get a fluorescence spectrum including the dark resonances, as shown in Fig. 6.3 (b) for
a three-level system, F has to be calculated for every detuning ∆i of the corresponding
light field within the range of interest.
6.2.2. Dark resonances and thermal motion
Now, we take the finite temperature of the ion into account, assuming a thermal state at
temperature T. For conditions with phonon numbers n ≫ 1, the motional state of the
87
6. Incoherent thermal excitation
Figure 6.4. – (a) Zeeman splitting lifts the degeneracy of the three-level system (dashed lines), leading
to a 8-level system, not drawn to scale. The polarization of the two light fields leads to 4
possible coherent transitions from the ground states S1/2 into the metastable states D3/2. (b)
Calculated spectrum of the 8-level system under respect of the given polarizations, leading to
4 dark resonances. The background colors indicate the corresponding transitions.
ion can be introduced in a semi-classical way. We consider the two light fields to have an
angle α with respect to each other. A velocity ~v of the ion results in a Doppler shift of the
two frequencies ωa and ωb and thus modifies their detuning by the vector product
δ∆i(v,α) =~ki ·~v, (6.18)
where~ki are the corresponding wave-vectors. Through the inclusion of the Doppler
shifted detunings in the calculation of the spectrum we get a fluorescence rate F (α,~v)
depending on α and ~v. We obtain a temperature-dependent fluorescence rate F̃ (T)
∫
F̃ (T) = PT(~v) · Fα(~v) d~v, (6.19)
by integrating over all velocity classes in the Maxwell-Boltzmann distribution
( )
m 3/2
P (v) = e(−m~v
2/2k
~ B
T)
T (6.20)2πkBT
with atomic mass m and Boltzmann-constant kB.
For a numerical solution of Eq. (6.19) we discretize ~v into distinct velocity classes
along the directions of the laser beams. To obtain a fluorescence spectrum, scattered
photons near 397 nm are recorded as a function of the detuning ∆866, while ∆397 is kept
constant. Note, that scanning ∆397 is unfavorable for thermometry, as its detuning has
a strong influence on the cooling rate. As a magnetic field is applied to define the
quantization axis, the degeneracy of the electronic states is lifted, resulting in an 8-level
system [Sch95, Obe99], see Fig. 6.4 (a). Depending on the polarizations of the laser beams
up to 12 dark resonance lines can be observed. In our experiment the polarizations of
both beams are chosen such that only σ+ and σ− transitions are addressed, leading to
the appearance of 4 dark resonances (Fig. 6.4). Regarding the P1/2 ↔ D3/2 transition, the
σ±-polarization avoids dark states in the D3/2 manifold, which would not be coupled
to any P1/2 level. For the S1/2 ↔ P1/2 transition a σ±-polarization is convenient as it
leads to an excitation of transitions which are maximally apart and therefore most easily
resolved.
88
6.2. Principles of dark resonance thermometry
Figure 6.5. – 3D surface plot of the calculated fluorescence spectrum as a function of the frequency
detuning ∆866 and as a function of the temperature of the ions. The decreasing depths of
the resonances for increasing temperatures is emphasized by the contour plot on the bottom,
where blue (red) indicates low (high) fluorescence.
For the 8-level system, the matrix form of H as well as the coupling operators Ci are
given explicitly in [Tol14]. The Zeeman-splitting leads to a 64-dimensional Lindblad
operator Lik in Eq. (6.16) and thus to a time consuming matrix decomposition. To reduce
the computational complexity we approximate the effect of temperature on the dark
resonances by a relative Doppler broadening ΓD(T,α). It can be deduced from the
variance of the relative Doppler shift δ∆ = δ∆397 − δ∆866 and reads
∣ ∣√ √∣ √
ΓD(T,α) = ∣
k T
~k −~k ∣∣ B = k2 + k2 kBT397 866 397 866 − 2 k397k866 cos (α) . (6.21)2m 2m
Regarding the shape of the dark resonances, the laser linewidths and the Doppler broad-
ening act in a similar way. Therefore, ΓD is treated as an additional contribution to the
broadening due to the spectral widths of the lasers. During the following applications
both methods, using Eq. (6.19) and Eq. (6.21), have been compared for different spec-
tra and lead to consistent results for the temperatures within their uncertainties. The
resulting line shape of the dark resonances as a function of the temperature T is shown
in Fig. 6.5. To determine the ion temperature the measured fluorescence data is fitted by
F̃ (T) using a Markov-chain Monte-Carlo parameter estimation [Pos14].
The thermal broadening ΓD(T,α) shows a strong dependence on the relative orientation
α of the k-vectors of the two lasers. The strongest influence of the temperature, and thus
the highest sensitivity especially at low temperatures, is achieved for counter propagating
laser beams (α = π), where ∆397 and ∆866 are Doppler shifted with opposite signs.
However, already for moderate temperatures of a few millikelvin the contrast of the dark
resonances becomes too small to assign a reliable temperature. In this range, the best
89
6. Incoherent thermal excitation
choice is a collinear beam configuration (α = 0), which shows a reduced sensitivity, but is
well suited for a wide range of temperatures up to several tens of millikelvin, as shown
in Fig. 6.5. As most of the measurements in this thesis are performed at relatively high
temperatures, we will only use the collinear beam configuration here. A detailed analysis
of the sensitivity as a function of α, as well as measurements in the counter-propagating
beam configuration, are presented in [Tol14].
6.2.3. Shaping the laser cooling by exploiting dark resonances
For temperature measurements in general, the interaction of any measurement device
with the system under investigation changes the temperature of the system. Here, the
non-linear modification of the photon scattering rate in vicinity of dark resonances affects
the equilibrium temperature of the ion. However, this effect can be exploited to tune the
temperature of the ion only by using the non-linear properties of dark resonances. In the
following it is analyzed in a semi-classical picture, where the Doppler shift reduces the
light scattering of selected velocity classes.
For Doppler cooling, during which the 866 nm laser may be far detuned from dark
resonances, the cooling rate is limited by the spontaneous emission lifetime τP = 6.9 ns
of the P1/2 level. We assume the weak binding regime, which means that the period
of the secular motion is much larger than the lifetime, ω ≪ 1/τP. The steady state of
the internal dynamics is typically reached after 10−7 s and therefore, at least in our case,
much faster than any secular motion. The ion does not move considerably during the
scattering process [Ste86b, Met99] and we can assume a constant velocity. In the vicinity
of a dark resonance the velocity-dependent Doppler cooling force is affected by the
modified scattering rate. To analyze its influence we introduce the effective Doppler shift
δ∆DR of the relative detuning of the laser frequencies with respect to a dark resonance,
∆DR = ∆866 − ∆397. In consideration of δ∆DR, the condition for a dark resonance reads
∆DR = ∆DR − δ∆DRδ = ∆866 −
~v ·~e
∆397 − (ν866 − ν397) = 0, (6.22)c
where νi are the absolute frequencies of the lasers,~e is their direction and ~v the velocity
vector of the ion. The relative Doppler shift δ∆DR has the opposite sign of the two
individual Doppler shifts, because ν866 < ν397. If the ion moves in the same direction
as the laser beams, ~v ↑↑ ~ki, photon scattering leads to an amplification of the motion
of the ion. This heating process limits the achievable temperature during Doppler
cooling [Met99]. However, if the frequency of the 866 nm laser is red detunedwith respect
to the dark resonance (∆DR < 0), those velocity classes with ~v ↑↑~ki are Doppler shifted
into resonance ∆DRδ ≈ 0. Photon scattering then is suppressed and the resulting excitation
of the ion motion is reduced. As a consequence, for ∆DR < 0 a lower temperature can
be achieved, similar to the velocity dependent coherent population trapping (VSCPT)
schemes for samples of neutral atoms [Asp88].
In Sec. 6.6 we present the corresponding measurements which demonstrate how the
non-linear features of the scattering rate of photons in the region of dark resonances are
used to tune the temperature of the ions within a large range.
90
6.3. Determination of the laser linewidth
6.3. Determination of the laser linewidth
To obtain the temperature from measured dark resonances, a precise knowledge of
the laser linewidths δνi is crucial. There are several common methods to determine
the linewidth of continuous-wave lasers. In general, the determination of an optical
frequency requires a frequency reference, which provides a higher accuracy than the
linewidth under investigation. A beat signal of both frequencies then allows to convert
fluctuations from the optical regime to frequencies which can be evaluated with elec-
tronics. For broadband lasers (e.g. ∆ν > 1GHz, multimode operation) techniques of
spatial spectroscopy are sufficient, such as diffraction patterns of optical gratings. For
narrow bandwidth lasers (∆ν < 100MHz, single-mode operation) a high-finesse optical
resonator can be used as a frequency reference. However, to construct a resonator with
such a narrow linewidth is a complex and costly affair. Less demanding is the common
technique of a self-heterodyne linewidth measurement, where the spectral line of the
laser under investigation is used as frequency reference at the same time [Oka80]. The
beam therefore is superposed with a delayed portion of itself, and the spectral width
of the beat note reveals the laser linewidth. However, for narrow bandwidth lasers the
coherence length is on the order of 104m, and the required delay lines are expensive and
bulky.
Here we show that an external locking-cavity can be used to measure ∆ν, although the
cavity linewidth is much broader than the laser linewidth. We exploit the error-signal,
resulting from the Pound-Drever-Hall lock, to convert frequency fluctuations of the laser
to voltage fluctuations, as illustrated in Fig. 6.6. The conversion coefficient is defined
by the slope of the error-signal around its zero-crossing. The error-signal is obtained
from a scan of the laser frequency over the resonance of the cavity, and, as a consequence,
the recorded error-signal is a function of time. In order to determine the velocity of
the scan in units of the laser frequency, we set the range of the scan to more than a
full spectral range (FSR) of the cavity, as demonstrated in Fig. 6.6 (b). The resulting
conversion coefficient allows us to derive the spectral linewidth of the laser from the
variance of the locking signal, while the laser is locked to the cavity, see Fig. 6.6 (c). By
this procedure, the σ-linewidths of the 866 nm laser and of the 397 nm laser result to
δν866 = 209(12) kHz (6.23)
δν397 = 173(8) kHz. (6.24)
Here we have subtracted the electrical background noise quadratically, which has been
determined in a separate measurement. The linewidths are in the specified range of
δν < 300 kHz for the types of grating stabilized diodes which are used [DK14]. As
expected, the emission spectrum of the 397 nm laser is narrower than the one of the
866 nm laser, which results from the additional frequency filtering by the SHG cavity.
The precise knowledge of the laser linewidths is the basis for the determination of an
absolute temperature in the following sections, and, at the same time, limits the precision
of the temperature measurements.
91
6. Incoherent thermal excitation
Figure 6.6. – Procedure to determine the linewidth of the laser sources. Here, as example, the
linewidth of the 397 nm laser. (a) A scan of the full spectral range of the reference cavity is
used to calibrate the scan velocity. The graph shows the transmission signal (blue) and the
error signal of the cavity (orange). (b) Error signal from the PDH-lock (black). The slope of
the linear control signal around the zero-crossing (orange) is measured to 22.2(2)mV/MHz.
When locked to the zero crossing, a frequency oscillation (blue) is linearly translated to a
voltage oscillation. (c) Resulting voltage fluctuations of the error signal of the locked laser
(blue line). The σ-width of the distribution of the fluctuations (blue bars and orange fit) results
to σ397 = 4.36mV and reveals a laser linewidth of ∆ν397 = 173kHz.
92
6.4. Temperature measurements in thermal equilibrium
6.4. Temperature measurements in thermal equilibrium
6.4.1. Experimental parameters and characteristics
For the measurements in this section we drive the trap electrodes with a peak-to-peak
voltage of about 890V, leading to radial trap frequencies of ωx/2π = 468 kHz and
ωy/2π = 472 kHz. Applying a constant voltage of 9V to both end-caps results in an
axial trap frequency of ωz/2π = 170 kHz.
We use beam path B1 from Fig. 4.10, where the two laser beams are oriented parallel
to one another and perpendicular to the magnetic field. The laser beams address and
cool all vibrational modes simultaneously. The polarization of both beams is chosen by
waveplates to be perpendicular to the magnetic field, leading to a superposition of σ+
and σ− polarization with respect to quantization axis. The intensity of the 397 nm beams
is 35 µW at a beam waist of 160 µm and for the 866 nm laser 80 µW at a beam waist of
380 µm. To record the fluorescence spectra the PMT is employed, while the camera image
is used for trapping, controlling and preparing the ions. The use of the PMT allows to
count single photons, which enables us to perform measurements with exposure times of
as short as some microseconds. For the spectral scans a constant intensity of the 866 nm
laser beam over a range of 50MHz is required. To avoid power fluctuations during
frequency scans, we employ the stabilization technique presented in Sec. 4.3.3.
6.4.2. Experimental procedure
To record a spectrum, the frequency of the 866 nm beam is scanned via the drive frequency
of the according AOM. We can chose two different modes of operation: First, a wide
scan of the spectrum including all dark resonances is recorded, where the frequency is
tuned continuously over the region of interest and the scattered photons are recorded.
The corresponding spectrum is used to obtain most of the external parameters, which are
assumed to stay constant subsequently. A fit of the data by the calculated fluorescence
rate F (∆866) allows to determine Rabi-frequencies Ω397 and Ω866, detuning ∆397, and
the magnetic field strength B.
The second mode of data acquisition is a pulsed operation, where we only scan over
one of the resonances. The temperature is determined from the model fit as the only free
parameter. As discussed before, scattering photons within dark resonances affects the
temperature of the ion. In order to greatly reduce these processes, we perform scans as
follows:
i) For an equilibration time te ≥ 2ms the frequency of the 866 nm laser is set at some
fixed detuning ∆e, far from any dark resonance, resulting in a steady-state of the ion
temperature.
ii) Then, for a much shorter measurement time of typically tm = 20 µs, the drive
frequency of the AOM is switched to a detuning ∆m by switching between the two VCO
drivers of the AOM, see Fig. 4.9 and App. A.6. Only during tm the fluorescence photons
are recorded on the PMT.
This procedure is repeated multiple times for detecting in total about 104 photons at
the desired ∆m. Then, ∆m is stepped to the next value to obtain the full shape of one of
the dark resonances. A fit with the analytical model reveals the temperature T of the ion.
93
6. Incoherent thermal excitation
Figure 6.7. – (a) Dark resonances for different applied electronic noise amplitudes (0Vrms (black),
300mVrms (blue), 800mVrms (red)). A fit yields Ω397/2π = 12MHz, Ω866/2π = 8MHz,
∆397/2π = 14MHz and B = 4.7× 10−4 T. The effective temperatures result to 3.1(5)mK
(black), 9(1)mK (blue) and 78(6)mK (red). (b) Measured temperature as a function of the
noise amplitude, following T ∝ N 2.
6.4.3. Measurement of steady-state temperatures
We now demonstrate temperature measurements with a linear crystal of 5 ions. The
number of ions increases the heat capacity of the crystal because of a larger number
of vibrational modes, but the steady state temperature, being an intensive quantity, is
the same as for a single ion. On the other hand, using more ions increases the rate of
scattered photons and therefore reduces the required number of repetitions for each
measurement.
Fig. 6.7 shows the recorded fluorescence spectra at different amplitudes of the electric
field noise. The temperature is obtained by a fit of F (∆866) to the data. In absence of
any heating process we measure the experimental cooling limit to be T0 = 3.1(5)mK.
As compared to the theoretical limit of Doppler cooling for the 40Ca+-ion, Eq. (2.22),
TD = 0.55mK is still a factor of of 6 lower than T0. However, TD is based on calculations
for a 2-level system and for perfect cooling conditions. The experimental value T0
depends on the laser parameters as well as the trap geometry. As a side remark we
want to emphasize that our setup has not been optimized for reaching a small T0 but to
guarantee stable trapping conditions even for high noise amplitudes. In order to generate
higher temperatures of the ion we apply noise amplitudes of 300mVrms, 500mVrms and
800mVrms. The fit to the data reveals temperatures of T = 9(1)mK, 30(3)mK and
78(6)mK, respectively. Thus, the temperature shows the expected quadratic dependence
on the electric field noise amplitude T = T0 + αN 2, with T0 = 3.1mK, see Fig. 6.7 (b). A
fit to the data reveals a temperature dependence of α = 110mK/V2.
For the frequency scans presented in Fig. 6.7 we have used the continuous acquisition
scheme only. Interestingly, at low temperatures, we observe deviations from the fit curve
on the high energy side of the resonances, most pronounced close to ∆866 = −6MHz. At
the same time we observe that the dark resonances appear to be deeper than expected by
the analytical model. These peculiarities are a result of the heating and cooling effect in
vicinity of the dark resonances, as predicted in Sec.6.2.3. Later, in Sec. 6.6 we will exploit
94
6.5. Investigation of thermal dynamics
Figure 6.8. – Scan of the |S1/2,− 1/2〉 − |D3/2,3/2〉 dark resonance in the pulsed acquisition mode.
The frequency of the 866 nm laser is set to ∆e for an equilibration time te ≥ 2ms. Then, for
tm = 20 µs, the frequency is switched to ∆m and the fluorescence is recorded. By scanning ∆m
the shape of the spectrum is obtained. A fit of F (T), with T beeing the only free parameter,
reveals the temperature of the ion. Here, we find T = 3.1mK for the blue resonance and
T = 9.4mK for the red resonance.
these effects to tune the temperature of the ions deliberately.
However, these artifacts in the recorded spectra are absent in the pulsed acquisition
mode. Fig. 6.8 shows the spectrum of the most pronounced resonance, which corre-
sponds to the |S1/2,− 1/2〉 − |D3/2,3/2〉 transition. The measurement time of tm = 20 µs
guarantees that the influence of the measurement itself on the temperature is negligible.
All free parameters of F (∆866) except the temperature have been determined by a scan
of the full spectrum. Then, measuring a single resonance in the pulsed acquisition mode
is sufficient to determine the temperature of the ion precisely.
6.5. Investigation of thermal dynamics
One of the major strengths of the dark resonance thermometry technique is the ability to
capture the time evolution of the temperature of the ion during thermal equilibration
processes. This feature is a mandatory prerequisite for the analysis of the reservoir
interaction of thermodynamicmachines and creates new possibilities for the investigation
of thermal dynamics in general.
To demonstrate such an equilibration measurement, we start with a laser cooled ion.
The laser cooling process mimics an interaction with a cold reservoir at TC. In order to
bring the system out of equilibrium we apply an electric noise pulse, corresponding to a
sudden temperature change of the reservoir. Consequently, we observe an increase of the
temperature T of the ion. We record fluorescence spectra with the pulsed measurement
technique at different time delays between 50 and 650 µs after the start of the noise
pulse, see Fig. 6.9 (a). Fitting the measured temperatures with a thermalization function,
TH(t) = TH (1− exp (−∆t/τH)), reveals a time constant of τH = 257(40) µs. Similarly,
we observe the inverse cooling process. Therefore we apply an electrical noise pulse of
2ms, while laser cooling the ion with a fixed detuning ∆e to reach an initial steady state
temperature TH . Then, we switch the noise off and observe the re-cooling process, see
95
6. Incoherent thermal excitation
Figure 6.9. – Time resolved thermal dynamics of trapped ions. (a) Cooling of an ion crystal: Ions
are first simultaneously excited by electrical noise and laser cooling (∆e = 0), resulting in a
steady state temperature of T = 49(5)mK as measured from the shape of dark resonances,
see inset. Once the electrical noise is switched off, the laser cooling reduces T with an
exponential constant τC = 87(2) µs, reaching finally TC = 3.0(5)mK. (b) Ion heating dynamics:
Starting with a laser cooled crystal, we turn on the electric noise, which increases T with
τH = 257(40) µs to a steady state at TH = 81(7)mK. In both cases we use short exposure
times of tm = 20 µs.
Fig. 6.9 (b). An exponential fit of TC(t) = TH exp (−∆t/τC) + TC reveals τC = 87(3) µs.
Here, the heating rate of the trap [Tol14, Bro15] is neglected as compared to the concurrent
errors: at low temperatures the dominant error results from the systematic uncertainty
of the spectral linewidth Γ of the laser sources. An uncertainty in the knowledge of
the laser linewidth of 30 kHz results in a temperature uncertainty of 0.4mK. For high
temperatures, the statistical error of the fits dominates.
In the language of reservoir engineering [Cir94b, Poy96], such heating and cooling
processes are interpreted as thermalization of the ion with an effective heat bath. The
thermal interaction of the ion with the cooling laser and the electric field noise has the
same characteristics as if the ion is coupled to a real thermal reservoir: thermalization to
a finite temperature (the bath temperature) and an exponential temperature evolution
with fixed time constant.
6.6. Engineering thermal states by exploiting dark
resonances
In addition to the temperature measurements presented above, the dark resonances
themselves can be used to adjust the temperature. As shown in Sec. 6.2.3, the modified
scattering of photons affects the properties of Doppler cooling. Here, we demonstrate
how the temperature can be controlled and measured in vicinity of the dark resonance at
∆DR0 corresponding to the |D3/2,−3/2〉 state, see Fig. 6.10. Again, we apply the pulsed
measurement procedure where the frequency of the 866 nm laser is switched. If ∆e
is close to the P1/2 ↔ D3/2 transition and far from any dark resonances during the
equilibration time te in step (i), we measure a temperature of T0 = 3.1(4)mK (as done
96
6.6. Engineering thermal states by exploiting dark resonances
10 a) b)
3.0
8
2.0
6
4
T0 1.0
2
TD
0.0
0
-5 0 5 10 1 10 100 1000 10000
Relative detuning Δ DRe - Δ0  (MHz) Measurement time tm (µs)
Figure 6.10. – Influence of the dark resonance on the temperature of the ion. (a) Measured equilib-
rium temperatures (blue points) while varying the detuning ∆e. On the low frequency side
of the resonance, T drops below T0 and we reach 0.7(4)mK (blue shaded area), still above
TD . On the high frequency side of the resonance, T rises above T0 (red shaded area) and
even diverges at ∆ DRe ≈ ∆0 + 2MHz. The gray resonance shape in the background serves
as orientation, at which point of the resonance feature the measurements have been taken.
(b) Influence of the measurement time tm on the resulting temperature. The difference of the
measured temperatures for ∆e = ∆DR0 + 8.5MHz and ∆ = ∆
DR
e 0 − 1.2MHz (green markers
in (a)) has been evaluated as a function of the measurement time tm. An exponential fit
∆T = a exp(tm/τm) reveals a time constant of τm = 0.38(7)ms.
before). However, if the frequency of the 866 nm laser is close to a dark resonance but
red detuned ∆ < ∆DRe 0 during te, we see that the temperature drops by a factor of four
and measure 0.7(4)mK at ∆ = ∆DRe 0 − 1.2MHz. In contrast, we observe strong heating
effects if ∆e is blue detuned to the dark resonance ∆e > ∆DR0 . At ∆e = ∆
DR
0 + 2MHz the
ion is excited so much that the dark resonance feature, recorded during step (ii), vanishes
within the statistical scattering. We can only give a lower bound of T > 80mK.
Because different frequencies along the dark resonance lead to different equilibrium
temperatures, we investigate experimentally the influence of the measurement duration
tm in step (ii) on the resulting temperature. Therefore, we vary tm between 2 µs and
1ms. In the latter case, tm is sufficiently long to let the thermal state equilibrate to a
temperature determined by ∆m during the measurement. For short times of tm, however,
the temperature is governed by ∆e and not significantly influenced by the measurement.
We don not use tm < 2 µs, since the internal dynamics need ≈ 0.5 ns to reach a steady
state of the occupation probabilities [Tol14]. To characterize the influence of tm, we have
tuned ∆e to two different values at ∆DR0 − 1.2MHz and ∆DR0 + 8.5MHz (green highlights
in Fig. 6.10 (a)), corresponding to temperatures of 0.7(4)mK and 3.1(4)mK, respectively.
The equilibration process is shown in Fig. 6.10 (b), where the measured temperature
difference ∆T between both points is plotted as a function of tm. An exponential fit
of the temperature differences by ∆T(tm) = a exp(−tm/τm) reveals a decay time of
τm = 0.38(7)ms. Indeed, for measurement times tm ≤ 20 µs the modification of the
measured temperature of ∆T ≤ 0.1mK is negligible as compared to the concurrent
uncertainties.
97
Temperature (mK)
Measured temperature difference (mK)
6. Incoherent thermal excitation
4.0
3.0
2.0
1.0
40 60 80 100 120 140 160
Trap frequency ωz/2π (kHz)
Figure 6.11. – Determination of the PSF of the imaging optics by recording the spot size ∆zcam
for different values of the trap frequency ωz. This exemplary measurement has be done
on the axial d√egree of freedom. The PSF is the offset for ωz → ∞ of the fitted function
∆zcam(ω 2 2z) = kBT/(mωz ) + ∆zPSF, while the temperature is extracted from its curvature.
6.7. Spatial thermometry technique
A different approach to determine the temperature of the trapped ion is an analysis of the
spatial spread of the wavefunction of the ion, or, more precisely, of the spatial fluorescence
distribution arising from the ion. An ion with higher thermal energy occupies a larger
volume inside the harmonic trapping potential. As shown by the group of Hänsch in
2012, an analysis of the increased spot size of the fluorescence distribution of hot ions
on camera images can be used to determine their temperature [Knü12]. While the
measurement of low temperatures is limited by the spatial resolution of the imaging
optics, this technique is well suited for T ≥ 100mK, where the dark-resonances are
not resolved anymore. In the classical weak-binding regime, kBT ≫ h̄ω, the root mean
square width of the Gaussian probability distribution of a thermal ion at temperature T
is √
k T
∆x Bth = 2 . (6.25)mωx
In order to extract ∆xth from the camera images, we have first to calibrate the resolution
of the imaging system, given by its point spread function (PSF) [Knü12]. The PSF is the
root mean square width of an image of a point source. Assuming an isotropic broadening,
the total width of the recorded fluorescence distribution reads
√
∆x = ∆x2cam PSF + ∆x
2
th. (6.26)
To determine ∆xPSF precisely, we make use of the dependance ∆x ∝ ω−1th x from Eq. (6.25)
and measure ∆xth for different values of ωx. Such a measurement is shown in Fig. 6.11,
where we assume the laser-cooled ion to have a constant temperature T0. The PSF
corresponds to the extrapolated spot size ∆xcam for ωx → ∞, where the thermal wave-
function of the ion is compressed to a point source. By fitting Eq. (6.25) to the data
we find ∆xPSF = 1.41(9) µm and T0 = 3.2(2)mK. Within their statistical uncertainties,
the resulting temperature T0 is in accordance with the measured temperature of a laser
98
Spot size Δzcam (µm) 
6.7. Spatial thermometry technique
Figure 6.12. – Temperature measurement using the spatial thermometry technique. (a) Projection in
radial direction of the recorded fluorescence distribution of a cold (blue) and a hot (orange)
ion, fitted with Gaussian probability distributions. The resulting temperatures are 4.7mK
and 1.28K. (b) Evaluation of the measured temperatures as a function of the electric noise
amplitude N , while the noise has been applied in a pulsed mode. A fit of the function
T(N ) = aN 2 + T0 demonstrates the accordance of the resulting temperatures with the
expected behavior.
cooled ion using the dark-resonance thermometry technique. This comparison serves as
a cross check for the validity of both methods.
Knowing the value of ∆xPSF allows us to determine the temperature of the ion directly
from the fluorescence distribution on camera images, if ∆xcam > ∆xPSF. An evaluation
of the temperature of the ion for different amplitudes of the electric noise N is shown
in Fig. 6.12 and confirms the expected quadratic dependance T ∝ N 2. Here we have
applied the noise in a pulsed mode, in order to cover a larger range of noise amplitudes:
we alternate between heating and cooling, whereby we apply heating for 4.1 µs and
cooling for 8.2 µs. In this way a thermal equilibrium is reached at some intermediate
temperature, which is then measured by spatial thermometry. Later, for the heat engine
operation in Sec. 7.4, we will identify the temperature values from Fig. 6.12 as working-
point temperatures Tw of the engine. Note that the technique presented in Fig. 6.11,
where the trapping potential is varied, cannot be employed for hot thermal states during
electric noise heating. The reason is that a variation of the spatial extent of the probability
distribution leads to a modified heating rate due to the quadratic nature of the trapping
potential. For more details and an elaborate discussion of systematic uncertainties of the
spatial thermometry, see [Knü12].
To summarize, the spatial thermometry technique is easy to use and is evaluated
with low effort. It is well suited for hot thermal states and complements seamlessly
the temperature range where dark resonance thermometry can be applied. With both
techniques at hand, we cover a temperature range from below 1mK up to 10K. This
coverage will be entirely used when we characterize the temperatures during heat engine
operation in Sec. 7.4. However, the spatial thermometry technique relies on a spatially
resolved imaging of the fluorescence distribution and therefore the use of a PMT is
impossible. This fact limits the time resolution of the temperature measurement to the
exposure time of the EMCCD camera and thus to several milliseconds. Therefore, the
spatial thermometry technique is appropriate for steady state temperatures and can not
be employed for the measurement of fast thermal dynamics. Additionally, the thermal
99
6. Incoherent thermal excitation
distribution must exceed considerably the resolution of the imaging system [Nor11].This
concern may be circumvented by using standing waves to probe the spatial extent of
the ion [Wal12a], however, involving a high technical effort to obtain laser beams with
interferometric stability.
6.8. Outlook and applications
In this chapter we have first shown, that by the combination of electric noise and laser
cooling, an effective and adjustable heat bath for trapped ions is emulated. We then
have adopted the dark resonance thermometry technique for the first time to measure
the resulting temperatures of trapped ions. As we will show in Sec. 7.4, only through
such a fast and almost non-invasive method we are able to measure and evaluate the
thermodynamic cycles of the single-ion heat engine. We have shown that dark resonances
in a multi-level electronic system, here in 40Ca+, are a versatile tool for controlling and
measuring the temperature of a single ion or a small ion crystal. We observe and interpret
the modification of the dark resonances and determine the temperature of the ions from
a fit of the laser-induced fluorescence data by a model of the expected lineshape. This
technique is convenient to use as it only requires the light sources which are used for
Doppler cooling and for exciting the ion’s fluorescence. Using this type of thermometry,
and having characterized the spectral linewidths of the laser sources, we are able to
measure temperatures between 0.7mK and 61mK.
Beside the application to thermal machines, we see advantages of the dark resonance
thermometry for applications with low trap frequencies or large ion crystals, where the
quantized energy structure can’t be resolved and thus sideband spectroscopy techniques
are not applicable. Our method may be well suited for studies of non-linear dynamics
in ion crystals, such as structural phase transitions in ion crystals and the formation
of defects [Ulm13]. Non-linear couplings may be studied in systems with many ions
and in the presence of thermal excitations with two-dimensional spectroscopy [Lem15],
in order to detect signatures of a structural phase transition of the ion crystal, as well
as resonant energy exchange between vibrational modes. Our method fills the gap
between the sideband spectroscopy technique, used for very low temperatures close
to the ground state of motion, and classical techniques, such as the Doppler recooling
method, which are well suited for hot thermal states with temperatures above 104mK.
Its temperature range overlaps with the one of the spatial thermometry, which has been
used to crosscheck the validity of both techniques. A combination of both extends the
accessible range for steady state temperatures to largely above 103mK.
We have demonstrated that a measurement of thermal dynamics can be performed
on the timescale of a few microseconds. This opens the door for future measurements
of thermal equilibration processes in non-equilibrium situations, such as time-resolved
transport of thermal energy through an ion chain or ion crystal [Lin11, Pru11, Ber13].
100
7 Experimental implementation of the
single-atom heat engine
In this chapter we demonstrate experimentally the working principles of the single-ion
heat engine. As discussed before, we use a combination of laser cooling and electric field
noise to engineer cold and hot reservoirs. By exposing the radial state of motion of the
ion alternately to these heat baths, a closed thermodynamic cycle is realized. We show
that the work generated by the engine is transferred to the axial mode of motion and is
stored in an increasing oscillation amplitude. The ion trap setup allows us to control and
to analyze the state of the ion as well as to tune its interaction with the thermal reservoirs
precisely. The thermodynamic cycle of the engine is established for various temperature
differences of the reservoirs and the corresponding output powers and efficiencies are
characterized. We compare both, power and efficiency, with the expected values from
an analytical model. The ability not only to generate work, but also to store it and to
transfer it to other oscillatory systems is a crucial step towards the implementation of
more complex thermodynamic systems.
We first give an overview of the actual experimental setup and parameters in Sec. 7.1
and Sec. 7.2, before we demonstrate the measurement technique to record the ion tra-
jectory with high resolution in space and time in section in Sec. 7.3. Then, in Sec. 7.4,
we show how to reconstruct the thermodynamic cycles from the recorded trajectories
and the thermal dynamics. This is followed by Sec. 7.5 and an evaluation of the output
power derived from the heat engine cycles, as well as directly from the ion trajectories.
Sec. 7.6 focuses on the corresponding efficiencies. The experimental results are combined
with an analytical model for the expected power and efficiency of the engine, based only
on the measured temperatures and trap frequencies. This model is found to be in good
agreement with the behavior of the measured values. In section 7.7 we show that we
can exclude any other sources of excitation leading to an oscillation of the ion but the
conversion from thermal energy difference into mechanical motion. We end this chapter
with a short summary of the results in Sec. 7.8.
The experimental realization of the single-ion heat engine and the experimental results
are presented in a condensed form in our publication:
A single-atom heat engine,
J. Roßnagel, S. Dawkins, K. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler and
K. Singer,
Science 352, 325 (2016).
101
7. Experimental implementation of the single-atom heat engine
x y
z
Figure 7.1. – Condensed sketch of the setup of the heat-engine experiment, composed of a single
trapped ion (green), lasers for cooling, dissipation and observation of the ion (blue), radio-
frequency electrodes in funnel geometry (red), end-caps (gold) and outer electrodes (gray).
The position of the ion is imaged on an ICCD camera as a function of time, revealing its
trajectory (screen). A balun transformer inverts the electric noise signal from the generator,
see Sec. 4.1.3. The opposing noise signals are supplied to the outer electrodes, in order to
generate electric field noise on the trapping potential without affecting the trap frequencies.
7.1. Specific experimental parameters
Before we start to discuss the heat engine experiments we summarize the external
parameters for these measurements. The geometry of the trap has been presented in
Sec. 4.1. The funnel-electrodes are driven at a symmetrical radio-frequency voltage of
830V peak-to-peak at Ωrf/2π = 21.41MHz. This results in trap frequencies in radial
direction of ω0x/2π = 447 kHz and ω0y/2π = 450 kHz, at the axial equilibrium position
of the ion z = 0. The degeneracy of ωx and ωy is lifted, but they are sufficiently close to
permit an approximation of cylindrical symmetry with r2 = x2 + y2 and a mean radial
trap frequency ωr for the following analytical calculations. The axial trap frequency is
kept at ωz/2π = 81.2 kHz. The axial confinement is low enough to obtain oscillation
amplitudes in the range of micrometers due to the heat engine drive, but still high
enough to guarantee for stable trapping conditions. The thermal bath interactions
are acting on the radial degrees of freedom, as discussed before. The electric noise is
fed to the outer electrodes and provides a root mean square amplitude in the range
(0 . . . 7)Vrms. The cooling laser at 397 nm, corresponding to the reservoir interactions,
points in radial direction with 45◦ to both radial components. The beam is collimated
with a Gaussian diameter of σr ≈ 160 µm. The intensity is kept constant at Ir = 85 µW,
while the detuning is stabilized at ∆ωr = −55MHz = −2.4 Γ from the S− P resonance,
where Γ = 1/(2πτ) is the natural linewidth. The pulsed dissipation laser is applied in
axial direction, collimated with σz ≈ 175 µm, Iz = 20 µW and ∆ωz = ∆ωr. It leads to
an average damping coefficient of γ = 480(140) s−1, determined from a series of step
response measurements, as presented in Sec. 5.3. Regarding the repumping laser at
866 nm, we use one single beam which addresses all degrees of freedom and which is
stabilized at resonance, saturating the D3/2 − P1/2 transition. A condensed sketch of the
experimental setup is given in Fig. 7.1, comprising all relevant parts of the experiment.
102
7.2. Electronic control of the experiment
Tabor  WW1074
Master
trigger axial laser
TTL 2ω out
Agilent 33250A d
Slave trigger
ω trigger cameraTTL out
Keithley 3390 
gating
Noise noise to trap
out
Time
Figure 7.2. – TTL circuit and timing of the heat-engine drive consisting of three function generators
(orange). The Tabor WW1074 serves as master clock and oscillates at double the desired
driving frequency, 2ωz. It triggers the axial dissipation laser and the Agilent 33250A, which
serves as slave clock. The Agilent generates a TTL puls on every second trigger pulse, thus at
a frequency ωz, and with a variable delay d. The TTL signal from the Agilent in turn serves as
gating for the Keithley 3390, which generates the gated noise signal. The noise pulses are then
fed to the inversion circuit and to the trap electrodes.
7.2. Electronic control of the experiment
During the heat engine experiments, the TTLs for switching the axial cooling laser
and the electric field noise have been controlled by external function generators. We
could not use the common electronic control system of the experiment for the following
reason: when the control system is switching TTLs at any frequency, there is a current
modulation at the same frequency on the common ground line of the TTL control and
the DAC device. Due to residual resistances this leads to voltage oscillations in the entire
electronic system, and thus on the trap electrodes as well. The voltage modulation is too
small to be measured with the available devices, but large enough to excite resonantly
coherent oscillations of the ion. Such unintentional oscillations are easy to distinguish
from the heat engine drive, since the amplitude does not depend on the strength of
the heating and cooling processes and is still visible when the noise generator is not
connected to the setup. However, these oscillations conceal the signature of the heat
engine drive and thus must be avoided. Therefore, we have set up an independent TTL
management by a set of three coupled signal generators, shown in Fig. 7.2, with a power
supply decoupled from other devices. By this procedure we avoid to drive the oscillation
of the ion by coherent electric field kicks, which is shown and discussed in more detail in
Sec. 7.7.
7.3. Resolved measurement of the ion’s oscillation
In Sec. 2.5 we have shown analytically and numerically that a periodic heating and
cooling of the radial mode generates a coherent oscillation along the trap axis, if the
drive frequency matches the axial trap frequency. Here we demonstrate this fundamental
103
Voltage signal
7. Experimental implementation of the single-atom heat engine
Figure 7.3. – Alternating heating and cooling phases (red and blue shaded areas) lead to an oscillation
of the ion. The temperature changes induce a periodic driving force Fz (orange line) according
to Eq. (2.21), shown relative to its mean value of F0 = 5.03× 10−21 Nwhich stems from the
elevated mean temperature. The precise position z of the ion (black points) is determined
from the average of more than 200 000 camera images at each phase step. The error bars
result from the uncertainty of Gaussian fits to the recorded fluorescence images, see Fig. 4.8.
The measured positions are described by a sinusoidal fit (blue line). Approximating Fz by a
sinusoidal signal (black line) reveals a relative phase shift of 88(3)◦, in accordance with the
expected π/2 phase shift for a resonantly driven oscillator.
principle of our heat engine experimentally. Therefore we heat and cool the ion alternately
and tune the cycle duration precisely to tcyc = 2π/ωz = 12.32 µs. The engine therefore
resonantly drives a harmonic oscillation in axial direction. The duration of the heating
period is set to tH = 4.1 µs, resulting in a duty cycle of dH = tH/tcyc = 1/3. The cooling
period covers the rest of the cycle time, tC = tcyc − tH . The combination of heating
and cooling periods give rise to a closed thermodynamic cycle and leads to a periodic
force Fz(t) ∝ T(t) in the axial direction, as derived in Sec. 2.5. The time evolution of
the driving force Fz(t) is shown in Fig. 7.3 and provides a triangular-like waveform,
since tH ≪ τH and tC ≪ τC, with τ being the timescale of the respective thermalization
process, see Sec. 6.5. Fz(t) can be approximated by its first Fourier component, being a
sinusoidal signal, which then drives a coherent oscillation of the ion. A measurement
of the resulting oscillation of the ion is presented in Fig. 7.3. The measurement of the
resolved coherent oscillation demonstrates the transformation of thermal energy into a
mechanical motion, and thus manifests the functionality of the fundamental principles
of the heat engine system.
The work produced in each cycle is resonantly transferred to the axial degree of free-
dom and stored in the amplitude of the oscillation. The axial motion thus plays a role
similar to the flywheel of a mechanical engine. However, the essentially frictionless
nature of the system leads to an ever-increasing oscillation. In order to contain this
oscillation, we provide an artificial friction by introducing an additional pulsed dissipa-
tion laser in axial direction, as characterized in Sec. 5.3. The relative pulse length td of
the dissipation laser is set to td/tcyc = 0.1. Steady-state operation is reached when the
work generated by the engine is balanced by the energy dissipated by the damping. The
steady-state oscillation amplitude is analyzed in the following to characterize the output
power of the engine.
104
7.4. The thermodynamic cycles
To resolve phase and amplitude of the oscillation during the heat engine operation we
employ the ICCD camera, as described in Fig. 4.8. While the amplitude of the oscillation
can be recorded with a normal EMCCD camera, the fast gating and short exposure of
the ICCD is mandatory for a detailed reconstruction and analysis of the heat engine
cycles. To this goal, the camera takes images of the fluorescence of the ion at different
phases of the periodic heat engine drive. The exposure time is set to 700 ns, which is
18 times shorter than the axial oscillation period. The camera is synchronized with the
temperature modulation, allowing for the repetitive recording of images at particular
phases of the resulting oscillation. More then 200 000 images are integrated at a each
phase in order to collect enough photons for a reasonable analysis of the ion’s position.
The fluorescence distribution of the 2-dimensional camera images is projected on the trap
axis, and fitting a normal distribution allows to determine the position of the ion precisely.
By repeating this procedure at different phases of the driving signal, the oscillation of
the ion is reconstructed accurately, revealing the amplitude and phase of the resulting
sinusoidal oscillation, see Fig. 7.3. In order to characterize the heat engine properties,
those measurements have been done for different noise amplitudes, different relative
heating and cooling times and different driving frequencies.
For evaluating the thermodynamic cycles, a precise determination of the relative
timing is crucial. By fitting a sinusoidal signal to the measured oscillations we obtain
a delay of the ion’s trajectory to its expected oscillation, which results from the gating
for the heating pulses and the corresponding force. The measured average delay is
d f = 1500(1100) ns, without showing a systematic dependency on any variation of
external parameters. This delay is found to have two major contributions. One is an
electronic delay of the noise pulse on the trap electrodes with respect to the gating pulse.
It results from charging effects of the low pass filters, the transformers and the bias
tees and has been measured to 800(100) ns. The second contribution results from the
dynamics of the ion. The electric field noise modifies the velocity of the ion, not its
position directly. But the axial force component is sensitive to the position of the ion.
Hence, the force rises only with a delay of ωr/4 after the onset of the noise, leading
to another delay of 560 ns. The delay d f has to be respected for the timing of the axial
dissipation laser as well as for the following analysis of the heat engine cycles, and is
included in the relative timing in Fig. 7.3.
7.4. The thermodynamic cycles
7.4.1. Measurement of the thermodynamic cycles
Now, we want to reconstruct the thermodynamic cycles from the measured oscillation
data. In order to facilitate the evaluation of the cycles in terms of power and efficiency,
we here choose to present the mean phonon number 〈nr〉 = kBT/h̄ωr of the thermal state
in radial direction as a function of the corresponding radial trap frequency ωr [Lin03].
The latter can be obtained directly from the measured axial positions z(t) in conjunction
with the calibration measurements of the trap frequency ωr as a function of z, see Fig. 5.3
and Eq. (5.6). Regarding the small amplitudes of the oscillation ωr(z), Eq. (2.17) can be
approximated linearly, revealing
ωr(z)/2π = (−114(1)Hz/µm) · z+ 447.9(2) kHz, (7.1)
105
7. Experimental implementation of the single-atom heat engine
Figure 7.4. – Temperature dynamics of the ion in contact with the thermal reservoirs. (a) Thermal-
ization curves, derived from dark-state thermometry of the radial thermal state of the ion
when heated to TH (red) and cooled to TC (blue). Individual errors result from fits to the
dark-resonances as well as the uncertainty due to the laser-linewidths, as described in Sec. 6.5.
Individual fits reveal thermalization time constants for heating and cooling of τH = 189(26) µs
and τC = 216(13) µs, respectively. (b) Dependence of the hot bath temperature TH on the
root-mean-square of the electric noise applied to the trap electrodes, measured by dark-state
thermometry (circles) and spatial thermometry (triangles).
where the positive z-axis is defined in direction of the wide side of the funnel.
The temperature of the ion at any given moment of the cycle is deduced from the inter-
play of individually measured heating and cooling rates, determined by dark resonance
thermometry, as demonstrated in Sec. 6.5. The measurements of the heating and cooling
rates for the heat engine operation are presented in Fig. 7.4 (a). Individual fits to the
data reveal thermalization time constants for heating and cooling of τH = 189(26) µs and
τC = 216(13) µs, respectively. Both time constants only depend on the parameters of the
cooling laser, which are kept constant during the experiments. While the temperature
of the cold reservoir TC is set by the Doppler-cooling temperature T0, the temperature
of the hot reservoir TH is adjusted by changing the amplitude of the electric field noise.
Fig. 7.4 (b) shows the quadratic dependance of TH after thermalization as a function of
the root mean square amplitude of the electric noise N applied to the electrodes, see
Eq. (6.11). In order to cover a large range of accessible temperatures for this characteriza-
tion measurement we combined dark-resonance thermometry (see Sec. 6.4.3) and spatial
thermometry (see Sec. 6.7). However, heating the ion continuously with more than 2Vrms
noise amplitude leads to high loss rates of the ion. Nevertheless, to characterize the
temperature for higher noise amplitudes we apply alternately cooling and heating, as it
is done in the heat engine operation. From the measured steady state temperatures we
then calculate TH using the thermal dynamics from Fig. 7.4 (using Eq. (7.10), which will
be derived in the course of this section). This procedure opens an accessible temperature
range from 1mK to 4K. The fit shown in Fig. 7.4 (b) results in
TH(N ) = 3.4(1)mK+ 97(5)mK/V2 · N 2. (7.2)
The set of parameters comprising τH , τC, TH and TC fully characterizes the thermal
dynamics of the heat-engine cycle. We now have to apply the timings for the heating
106
7.4. The thermodynamic cycles
Figure 7.5. – Simulated temperature dynamics of the ion as a function of time during heat engine
operation using the measured parameters for thermalization from Fig. 7.4. Heating and
cooling durations in each cycle are tH = 4.1 µs and tC = 8.2 µs, respectively. With an
electric noise amplitude of 4Vrms, we determine a working point temperature Tw = 568mK
(black dashed line) and a temperature difference ∆T = T2 − T1 = 21.5mK (see inset). Tw
exp
is in excellent agreement with the directly measured value Tw = 570(40)mK (green bar),
presented in Fig. 6.12.
and cooling period in order to simulate the thermal dynamics of the ion. The duration of
one heat engine cycle is defined by the axial trap frequency, tcyc = 2π/ωz . The hot bath
interaction time tH is given by its duty cycle dH = tH/tcyc. Since the ion is in permanent
bath contact, the cold bath interaction time results to tC = tcyc − tH .
In Fig. 7.5, we present a simulation of the thermal dynamics T(t) of the heat engine
based on the parameters τH , τC, TH and TC, starting with an ion cooled to the laser
cooling limit T0. In the beginning, the heating process is much more efficient than
cooling, due to its steep slope for low temperatures, see Fig. 7.4 (a), and thus the mean
temperature increases. With increasing temperature the cooling process becomes more
and more effective, until it balances the heating and a steady state is reached, with a
mean working-point temperature Tw. The time evolution of the temperature of the ions
radial state of motion in steady-state operation is thus fully characterized by Tw and the
temperature difference ∆T = T2 − T1, which are the basis for the derivation of the heat
engine cycles. In addition to the simulations, the working-point temperatures have been
measured directly by spatial thermometry, see Fig. 6.12, and show an excellent agreement
with the simulated Tw (green bar in Fig. 7.5). Note that the heating and cooling processes
are much slower than the internal dynamics of the ion and thus can be regarded as
quasistatic with negligible losses due to irreversible processes [Aga13].
Here, we want to emphasize the need for resonant switching of the heating and
cooling processes. In addition to the fundamental mechanism that the resonant coupling
allows to extract the work produced and to store it in a separate mode, it is required
to allow for an effective detection of the heat engine operation. For instance, switching
a noise amplitude of N = 4Vrms leads to a typical increase of the temperatures from
T1 ≈ 562mK to T2 ≈ 584mK, as it is the situation in Fig. 7.5. By employing Fz(T) from
107
7. Experimental implementation of the single-atom heat engine
Eq. (2.21) and using the specific geometry of the trap, we calculate a corresponding
increase of the force in axial direction to be Fz = 2× 10−22N. Using Newtons law of
motion this force leads to a static displacement of z = Fz/(mω2z) = 11 nm within the
given axial trapping potential. Such a small displacement is impossible to detect with our
imaging setup. However, driving the cycle resonantly with the axial trap frequency leads
oscillation amplitudes large enough to be observed in our setup, as described above.
The thermodynamic cycle of the engine is obtained by plotting the phonon number
〈nr(t)〉 = kBT(t)/h̄ωr of the thermal state of the ion in the radial direction as a function
of the corresponding trap frequency ωr(t) [Lin03]. 〈nr(t)〉 is derived from the measured
temperature dynamics and ωr(t) follows from the position of the ion on the camera
images. An exemplary cycle is shown in Fig. 7.6, which has been recorded at a noise
amplitude of 4.0Vrms, a cycle period of tcyc = 2π/ωz = 12.3 µs and a heating time of
tH = 4.1 µs. Thus, the temperature evolution corresponds to the situation in Fig. 7.5 and
the axial oscillation to the one in Fig. 7.3. The uncertainties of the phonon numbers for
each point of the cycle are obtained by a variation of all relevant input variables within
their individual uncertainties, namely TH , TC, τH , τC and d f . The resulting changes are
separated into a global shift of the center of the cycle, as well as relative changes of the
individual points. The error bars in Fig. 7.6 correspond to the relative variations of the
cycle points, since only these contribute to an uncertainty of the area of the cycle, and
therefore affect the resulting output power and efficiency. The error bars of the relative
trap frequencies are obtained in the same way, respecting the uncertainties of the position
measurements as well as of the gauge measurement ωr(z), Eq. (7.1).
7.4.2. Analytical derivation of the thermodynamic cycles
To evaluate different series of thermodynamic cycles in order to characterize the heat
engine parameters, it is more convenient to elaborate analytical expressions for the key
parameters Tw and ∆T. Therefore we make use of the fact, that in a steady state situation
the temperature changes during the heating and cooling processes balance out. More
formal, we start from the first law of thermodynamics
dU = δW + δQ (7.3)
a∫nd assume that there is no residual change in the internal energy after one cycle,
cyc dU = 0. Furthermore we assume that the amount of heat converted into mechanical
work is much smaller than the absolute amount of heat absorbed from the hot reservoir
δW ≪ δQH . After the evaluation of work and efficiency in Sec. 7.5 and Sec. 7.6, we will
see that these assumption are satisfied. We obtain the steady state condition for the heat
exchange
δQH + δQC = 0, (7.4)
which can be transferred to temperature changes by using the definition for heat, δQ =
C dT, with the heat capacity C. To transfer Eq. (7.4) to the specific situation in our
setup, we use the fact that the heating and cooling processes lead to a variation of the
temperature around the same working-point temperature Tw. Including the different
duration of the coupling to the hot and the cold reservoir, we find
∆TH + ∆TC = ṪH (TH(t) = Tw) · tH + ṪC (TC(t) = Tw) · tC = 0, (7.5)
108
7.4. The thermodynamic cycles
Figure 7.6. – Thermodynamic cycle of the engine for one radial direction: the change in the trap
frequencies ωr is deduced directly from the measured z-positions, indicated by the second
axis. The temperature T of the radial state of motion and thus the corresponding mean phonon
number 〈nr〉 is determined from separate measurements (see text in Sec. 7.4.1). The values of
ωr and 〈nr〉 are given with respect to the center of the cycle at ω0r/2π = 447.9(2) kHz and
〈n0r〉 = 26160(445). The shaded area enclosed by the cycle reflects the work performed by
the engine, where red and blue colors indicate heating and cooling periods, respectively. The
gray line is the calculated trajectory of the cycle, see text. The error bars are obtained from
a relative shift of the data points with respect to the center of the cycle, under a variation
of all relevant parameters within their uncertainties. The four pictograms in the corners
illustrate the different strokes of an idealized cycle. They indicate the thermal probability
distributions of the ion in between the trap electrodes, while the background colors signify
the bath coupling.
109
7. Experimental implementation of the single-atom heat engine
where we evaluate the time derivative of the temperature Ṫ (t) at the time t where
T (t) = Tw. The underlying assumption Ṫ (T (t) = Tw) = const is valid for small
temperature changes ∆T ≪ Tw, which is satisfied for our experimental parameters, as
shown in Fig. 7.5. In order to obtain Ṫ (T (t) = Tw) we express the first derivatives of
the heating and cooling dynamics
[ ( )]
TH(t) = TC + TH 1− exp −
t
(7.6)
( ) τH
T tC(t) = TC + TH exp − (7.7)τC
as functions of the time dependent temperature T (t) and find
Ṫ T (TH + TC)− TH(t)H( H(t)) = (7.8)τH
Ṫ TC(t)− TCC(TC(t)) = − . (7.9)τC
Including Eqs. (7.8) and (7.9) to Eq. (7.5) we find the analytical expression for the working
point temperature
T
T = T + Hw C t . (7.10)1+ τH CτC tH
In order to obtain the temperature difference ∆T we insert Eq. (7.10) in Eqs. (7.8) and (7.9)
( ) ( )
1
∆T = T2 − T1 = Tw + ṪH(Tw) · tH −
1
Tw + Ṫ
T
C(Tw) · tC = Hτ τ , (7.11)2 2 Ht + CH tC
where we used the short form Ṫ (Tw) ≡ Ṫ (T (t) = Tw). The analytic expressions for
Tw and ∆T enable us to deduce the phonon numbers 〈nr(t)〉 and to reconstruct the
thermodynamic cycles. The cycles now only depend on the temperature of the hot
reservoir TH , defined by the amplitude of the electric noise N , and the duty cycle of the
reservoir interactions. All other parameters only depend on the cooling laser and are
thus constant. As we will show in the next section, a variation of these two parameters
allows to analyze the behavior of the power and the efficiency of the engine.
7.4.3. A Stirling-like engine type
The situation for the experimental realization of the single-ion heat engine is different to
the one of the simulations presented in Sec. 3.1. There, we have investigated an Otto cycle,
where we used independent heating and cooling processes, separated by an isentropic
expansion, where the system is completely isolated from any thermal reservoir. This
is an idealized situation which could unfortunately not be realized experimentally. To
implement an Otto cycle means to switch the radial cooling laser on and off at a rate
corresponding to the axial trap frequency. This, however, leads to a periodic modulation
of the light force on the ion and hence excites axial oscillations, as shown in Sec. 5.1.1. To
avoid light induced oscillations during the heat engine operation, we keep the cooling
laser on for the entire experimental sequence. Thus, the ion is in permanent contact with
one of the reservoirs, and consequently the dynamics are similar to those of a Stirling
engine [Bli12]. The idealized Stirling cycle consists of two isochoric temperature changes
110
7.5. Output power of the engine
and two isothermal expansion and compression processes, as discussed in Sec. 2.2.2.
However, this idealized Stirling cycle is smeared out by the harmonic oscillation of the
ion and the fact that a full thermalization with the reservoirs is not reached [Bra15b]. This
makes it difficult to distinct between the four individual steps. The same effect arises for
thermodynamic cycles of macroscopic engines [Cen15]. However, at this point we want
to emphasize that, of course, it is interesting to compare our engine to well-known cycle
types, but that such a comparison is rather of an academic nature and not essential to
evaluate the parameters of the engine and to prove its functionality.
7.5. Output power of the engine
7.5.1. Stationary output power
We derive the power output during steady-state operation in three independent ways.
We first determine the power Pcyc = Wcyc/tcyc by evaluating the workWcyc as the area
of the cycles, ∮
Wcyc = h̄〈nr〉 dωr. (7.12)
This work is computed by a numerical integration of the datapoints i of the closed
polygon according to Green’s Theorem [Bra86],
1 N ( )
W = ∑ ωi · 〈ni+1〉 − 〈ni 〉 ·ωi+1cyc r r r r , (7.13)2 i=1
( ) (
where ωN+1,〈nN+1〉 = ω1
)
r r r ,〈n1r 〉 . The resulting Wcyc corresponds to the work per-
formed in one of the two radial directions. However, since the engine is driven in both
radial directions simultaneously, the total power equals to Pcyc = 2Wcyc/tcyc. To assess
the performance of the engine, our system has been driven by various temperature
differences ∆T. As mentioned above, ∆T can be tuned either by adjusting the reservoir
temperatures via N , or by the relative duration of the reservoir interaction, defined by
dH . We have used both options to vary ∆T and have covered a range from 0mK to 70mK.
The resulting data is shown as orange points in Fig. 7.7 as a function of N and dH , and
reaches up to 3.4(5)× 10−22 J/s.
We alternatively deduce the power directly from the measurement of the axial oscilla-
tion amplitudes Az, which range up to Az = 15 µm. The dissipated power of a damped
harmonic oscillator at steady state yields [Dem08]
Posc = γmω
2A2z z , (7.14)
with the damping parameter γ according to Sec. 5.3. In a steady state operation the
dissipated power equals the energy input by any driving source. Thus, Posc corresponds
to the power generated by the engine, which is converted into oscillation energy and
displayed as gray points in Fig. 7.7 (c).
Both methods, Pcyc and Posc, use the same oscillation data as input variables and
lead, in principle, to the same power output of the engine. But, however, they differ
fundamentally in their derivation. Pcyc is based on the thermodynamic parameters of the
engine, namely ∆T, derived from individual measurements. Posc, in contrast, describes
the power for any harmonic oscillator, ignoring the origin and the form of the driving
111
7. Experimental implementation of the single-atom heat engine
Figure 7.7. – Output power of the engine. (a) A variation of the output power is achieved by different
electric field noise amplitudes N . For each point the power is calculated from the are of the
corresponding thermodynamic cycles. (b) The power is changed by modifying the duty cycle
of the heating period, at constant noise amplitude of N = 5Vrms (triangles) and N = 6Vrms
(diamonds). The power is again derived from thermodynamic cycles. (c) Combined analysis
of the power of the engine as a function of the temperature differences ∆T. ∆T is calculated
from the variation of the electric field noise (see (a), orange circles) and from the duty cycle
of the heating periods (see (b), orange triangles and diamonds). These data are consistent
with the expected value from analytical calculations (blue line). The values resulting from
the measured cycles are complemented by data from a direct analysis of the amplitude of the
oscillation (gray).
112
7.5. Output power of the engine
force. The resulting power values for both methods are consistent with each other and
range in the order magnitude of 10−22 J/s, depending on ∆T. This result shows that
the power from a thermodynamic evaluation corresponds to the expected power for
a damped harmonic oscillator. It underpins the functionality of the heat engine, or, in
other words, that the origin of the oscillation indeed is the temperature variation due to
heating and cooling processes.
We further calculate analytically the engines output power to compare this model to
the measured values. Therefore, we first compute the expected ion trajectory, assuming
the expressions for a driven and weakly damped harmonic oscillator at resonance,
see Sec. 5.3. The driving force Fz(t) is calculated from Eq. (2.21), accounting for the
temperature variations of the ion. The triangular-like temperature function T(t), shown
in Fig. 7.5, leads to a triangular-like driving function Fz(t). Since the latter drives
a harmonic oscillator resonantly, we need its first harmonic approximation Fsin(t) to
analyze the dynamics of the driven oscillator, see Fig. 7.3. To obtain a harmonic driving
signal we calculate the first harmonic component of a discrete Fourier analysis for the
periodic temperature function T(t),
∫ ( ) ( )2 tcyc 2π sin(π d) 2π
Thar(t) = T(t) · sin t+ φ dt = 2 − 2 ∆T · sin t+ φ . (7.15)tcyc 0 tcyc π (d d ) tcyc
To simplify the calculations, we introduce a phase shift of φ = −dπ with respect to the
time axis in Fig. 7.3, in order to make sure that the contribution of the other quadrature
vanishes. Thar(t) is now applied to the expression of Fz(T) from Eq. (2.21) and, neglecting
the weak z-dependence, we obtain
( )
≈ 4 kB sin(π d) tan θ 2πFhar(t) 2 − 2 ∆T · sin t+ φ . (7.16)π (d d )r0 tcyc
The ion trajectory is deduced from Fhar(t) assuming a driven and weakly damped
harmonic oscillator at resonance, Eq. (5.3). Having calculated the expressions for the
force and the resulting motion of the ion allows us to compute the analytic expression
for the resulting work per cycle,
∫ ∫ tcyc dz(t)
Wana = Fhar(t) dz(t) = Fhar(t) dt, (7.17)
osc 0 dt
where the integration is performed over one cycle period tcyc. Thus, for the output power
we find
W 4 k2 sin2
( )
ana B (πd) tan
2 θ
P = = ∆T2
J
ana = 8.84× 10−20 · ∆T2. (7.18)tcyc mπ4γ(d2 − d)2 r20 sK2
Fig. 7.7 shows that the measured thermodynamic power Pcyc agrees well with the
analytic model Pana, which is based only on the temperature variations and the trap
geometry, and does not depend on the measured oscillation data. This demonstrates
that ∆T is expected to cause an oscillation which corresponds to the measured ampli-
tude. Thus, it shows that there is neither an additional loss of oscillation energy nor an
additional input of the same.
Regarding the two evaluation methods for the measured data, we observe a signif-
icantly higher scattering of the Posc data compared to the Pcyc data. Both results are
based on the amplitude Az of the measured oscillation. On the one hand, there is the
113
7. Experimental implementation of the single-atom heat engine
resulting power from the analysis of the cycles which depends on Az, translated to the
trap frequency ωr(z), and on the temperature T. Thus, statistical scattering of Az enters
linearly in the resulting power Pcyc ∝ Az. On the other hand, there is the power deduced
from the dynamics of a harmonic oscillator depending quadratically on the amplitude
P ∝ A2osc z , see Eq. (7.14), and thus provides a scattering twice as large compared to the
data from the cycles. The larger error bars of Posc stem from the large uncertainty of the
damping coefficient, see Sec. 5.3.
Here, we face the main difficulty during the experiments: due to leakage currents
between the radio-frequency electrodes and the end-caps, combined with a finite resis-
tance of the their coating and their electric connection with in total 30Ohm, we observe
fluctuating charges on the end-caps. This leads to a variation of the axial potential and,
consequently, of the trap frequency of about ± 200Hz on a 10-100 s timescale, also ob-
served as small shifts of the ion along the trap axis. During the experiments we corrected
for those frequency variations between the individual runs, but, frequency variations
faster than the repetition of experimental runs could not be corrected. As a consequence,
this leads to a variation of the measured amplitude due to a mismatch of the driving
frequency and the resonance frequency. This amplitude variation is then translated into
a variation of the output power of the engine, with a linear dependence for Pcyc, while
Posc exhibits a squared dependence.
Additionally, we observe a systematic deviation between Pcyc and Posc. This most
likely originates from a systematic error in the calculation of the damping coefficient.
However, the deviation of the two data-sets is in the range of their individual error-bars,
and hence they are still consistent with each other.
In order to classify the measured power we calculate the corresponding power-to-mass
ratio ρ of the working agent (the single Ca+ ion) and compare it to ordinary car engines.
The maximum measured power of the single-atom engine, P = 3.4× 10−22W, leads to
ρsa = 5 kW/kg. Assuming a car engine with 100 kW (136PS), a total cylinder capacity
of 1 l and a density of the gas of 30 kg/m3 [Gau10], we find ρcar = 3MW/kg. Taking
account for the fact that the typical temperature difference in a car engine is of the order
100K instead of 100mK, both values scale in the same order of magnitude. Even though
there is a ratio of 1 : 1024 particles. This comparison, of course, is not substantiated for a
quantitative evaluation. The result implies, however, that the power of the engine scales
approximately as the number of particles in the working medium.
7.5.2. Start of the engine
Complementary to the steady state operation, we analyze the start of the engine from
rest. Therefore, we initialize the ion in a cold thermal state by laser cooling all degrees
of freedom. Then, at t = 0, we start to apply the heating and cooling pulses as well as
the damping laser. The noise amplitude has been set to 5Vrms and the heating pulses
repeat themselves at the resonance frequency ωd = 81.2 kHz. At t = 0 the damping
force is zero and increases with the ions velocity, Fd ∝ v. Therefore, in the beginning the
energy balance is positive and the engine starts to oscillate with an increasing amplitude,
until Fd compensates the driving force and a steady state is reached. To measure this
effect, we repeat a sequence of initialization and the start of the engine, and take images
of the fluorescence distribution after a fixed delay ts with 500 µs exposure time. After
integrating 200 images, we evaluate the width of the recorded fluorescence distribution.
Fig. 7.8 shows the result for a series of different values of ts, revealing the starting
114
7.5. Output power of the engine
Figure 7.8. – Start of the engine from rest. The increasing amplitude of the oscillation is measured as
a function of time. Therefore the width of the fluorescence distribution on camera images is
evaluated, relative to the initial distribution of an ion at rest (orange points). These data are
corrected by the separately measured thermal broadening (gray points), displayed relatively
to their mean value. To measure the thermal influence, the engine is driven non-resonantly,
leading to an increasing temperature without exciting a coherent oscillation. The fit to the
amplitude data follows Eq. (5.8).
behavior of the engine. The functional behavior is similar to the one presented in Sec. 5.3,
in the context of the measurements of the damping coefficient, including a beat note
between the driving frequency and the resonance frequency. Here, we obtain a damping
coefficient of γ = 140(90) s−1, using a power of the damping laser of 10 µW. Comparing
with the results in Fig. 5.7, this value is in agreement with the characterization of the
damping coefficient. Furthermore, the characteristic rise time of τs = 7(4)ms indicates
that steady state is reached after approximately 560 engine cycles.
In addition to the coherent oscillation there is a thermal broadening of the probability
distribution in the axial degree of freedom, resulting from photon scattering events.
In order to separate both effects, we have done the same measurement as above, but
with a drive frequency out of resonance at ωd = 88 kHz. Thereby, we measure the
accompanying heating of the axial degree of freedom, without exciting any relevant
coherent oscillation. The data shows that the thermal effect leads to a relative contribution
of 0.1 compared to the coherent oscillation amplitude. The data in Fig. 7.8 is corrected by
the measured thermal background.
To evaluate the power of the engine, we translate the corrected amplitude A(t) in an
oscillation energy E 2 2o(t) = mω A(t) . By calculating the time derivative of the energy,
dEo/dt, we obtain a value for the output power of the engine from the slope of the
increasing energy. Taking the dissipated energy into account, dEd/dt = −2γmv2, the
power of the engine results to P = 4(2)× 10−23 J/s. The uncertainty of this measure-
ment is significantly larger than the power values obtained from steady state operation.
Although the result here is in the same order of magnitude than the steady state power
values, it deviates significantly by 3 standard deviations. We conclude that the measure-
ment of the start of the engine is a concise demonstration of its dynamics, but is not
sufficiently precise to analyze the engine characteristics quantitatively.
115
7. Experimental implementation of the single-atom heat engine
Figure 7.9. – Efficiency of the engine. (a) The T,S-cycle of the heat engine, corresponding to Fig. 7.6,
measured as temperature relative to 568mK versus entropy relative to 179.6× 10−24 J/K.
Both work and efficiency can be derived from integrating this cycle. The square (dotted
line) represents the equivalent Carnot cycle for the full range of parameters and shows the
case of the theoretical maximum efficiency. (b) Measured efficiency as a function of ∆T, by
varying the noise amplitude (circles) or the duty cycle (triangles), compared to the result of
the analytical calculation.
7.6. Efficiency of the engine
We further evaluate the efficiency of the engine, η = −Wc/QH , fro∫m the measured data
by determining the heat absorbed from the hot reservoir, QH = H TdS. Therefore we
transform the heat engine cycle from the {n,ω}-diagram to a temperature-entropy repre-
sentation. The temperature is obtained directly from the corresponding measurements,
as described above, while the entropy S of a thermal harmonic oscillator is calculated
according to [Gem04, Aga13]
[ ( )]
kBTS = kB 1+ ln . (7.19)h̄ωr
The resulting thermodynamic cycle T(S) is shown in Fig. 7.9 (a). As expected, an
integration over its area leads to the same output power as we obtained by the integrat-
ing over the {n,ω}-cycle. The {T,S}-cycle has a significant diagonal structure, which
originates from the fact that S depends strongly on ωr as well as on T. In the {T,S}-
coordinates, a comparison with the equivalent Carnot cycle with maximum efficiency
is obvious, since the latter shows a rectangular shape [Gem04], which is indicated in
Fig. 7.9 (a) by a dotted line. The difference of the measured data to the Carnot cycle
reveals a substantial scope for optimization in future experiments. The efficiencies
have been calculated for the same measurements as done before for the evaluation of
the output power. The resulting values are displayed in Fig. 7.9 (b) and reach up to
η = 0.28%.
In order to obtain an analytic expression for the efficiency, ηana, we first calculate the
heat absorbed by the engine from the analytic expressions for T(t) and ωz(t). The former
is obtained as above from Fig. 7.5, while the latter is calculated from Eq. (7.1), applying
the analytical trajectory zana(t) from Eq. (5.3), using the corresponding driving force
Fhar(t). The heat absorbed from the hot reservoir is given by
∫ τ
QH
H dS(t)
ana = T(t) dt, (7.20)
0 dt
116
7.6. Efficiency of the engine
where the integration is performed over the heating period and where we neglect vanish-
ing higher order terms. Employing the above analytical approach, we derive the expected
behavior of the efficiency to increase linearly with a raising temperature difference ∆T.
We compute the analytic expression for the efficiency to be
W 4 k sin2(πd) tan2ana θ
ηana = in =
B
− 2 ∆T = (0.041K
−1) · ∆T. (7.21)
Q 3ana mπ γωz(d2 d)2 r0
Fig. 7.9 compares the analytical function ηana with the measured efficiencies. The
expected linear behavior matches qualitatively the resulting data. Nonetheless, due to
the fact that there are only 5 to 6 datapoints during the interaction with the hot reservoir,
the resulting values for the heat and thus for the efficiency are scattering stronger than
their individual errors would suggest. We have decided not to include this effect to
the errorbars, which are again deduced from a variation of the individual parameters
within their uncertainties. Additionally, we observe that the measured data have a
systematic tendency to higher values as compared to the analytical model. This may
be explained by the frequency mismatch of the heat engine drive. While a resonator
driven at resonance oscillates with a π/2 phase shift with respect to the driving signal,
its phase shift varies for not well-matched frequencies (see Fig. 5.1). This fact leads to a
rotation of the thermodynamic cycle around its center. While the work is unaffected in
first approximation, the calculated amount of heat absorbed by the system may decrease.
As long as the frequency mismatch is not known precisely, this effect can not be corrected
and is reflected in an overestimation of the efficiencies.
A comparison of the measured data with the correspon√ding efficiency at maximum
power, given by the Curzon-Ahlborn formula ηCA = 1− T1/T2 = 1.9%, reveals that
the trap parameters do not correspond to the optimal point [Aba12]. As shown in Sec. 3.2,
in principle the ion trap engine is able to achieve the Curzon-Ahlborn condition Eq. 3.17
by adjusting the oscillation amplitude to the temperature differences and vice versa.
However, in the realistic experiment we are limited by different aspects, discussed in the
following.
First, the amplitude at steady state may be adjusted by tuning the strength of the
damping force, but it is hard to adjust for practical reasons. On the one hand, reducing
the amplitude below the extent of the thermal distribution of the ion makes it hard to
detect. On the other hand, increasing the amplitude leads to higher velocities of the ion
an hence requires shorter exposure times of the camera to avoid blurred images. This
would increase the duration of the experimental sequences drastically. Furthermore,
a larger amplitude requires larger diameters of the laser beams in order to hold the
assumption of a uniform intensity distribution over the whole ion trajectory.
Second, regarding the temperature difference we are limited by a certain heating rate
through the electric field noise, depending on the trap frequencies, the noise amplitude
and the cooling laser. Technically, in our setup, N = 10Vrms is the upper limit. However,
for N > 7Vrms we obtain unstable trapping conditions for the given trap frequencies.
Third, the amplitude and the resulting temperature difference are not decoupled. A
stronger heating requires a stronger cooling in axial direction which implies a larger
damping force. We summarize that with a high technical effort we may approach
the Curzon-Ahlborn efficiency and maximum power output. However, as we have
demonstrated the working principle of the engine successfully, a further optimization
does not imply new physical findings and thus does not justify the required effort.
117
7. Experimental implementation of the single-atom heat engine
Figure 7.10. – Position of the ion as a function of time, as a test for a phase flip under inversion of
the signal lines. (a) A coherent oscillation is excited by electric field kicks at the resonance
frequency. A inversion of the signal lines leads to a 180◦ phase flip. (b) The system is now
driven in heat engine operation. The resulting oscillation is due to the thermal excitation of
the ion and does not undergo any phase flip when the signal lines are inverted.
7.7. Exclusion of non-thermal excitation of the ion
In Sec. 7.5.1 we have demonstrated that the temperature variation in combination with
the tapered trap geometry is the only origin of the oscillation of the ion. Nonetheless, in
this section we want to proof these findings again by a different approach. Therefore, we
compare the behavior of the system during the heat-engine operation with the behavior
under an artificial excitation of the oscillation. As mentioned above, see Fig. 7.1, the
electric noise is fed with inverted sign to two opposing outer electrodes. A pulsed noise
signal modulates the temperature of the radial state of the ion, resonant with the axial
trap frequency. The heat-engine operation converts the temperature difference into a
coherent motion along the trap axis. But, assuming the noise signal has a small constant
offset in average, the pulsed signal leads to repetitive radial kicks to the ion, which
have constant phase and amplitude. Due to the taper of the trap, the radial field has a
contribution in axial direction and its modulation leads to a coherent oscillation of the
ion along the trap axis, independent from the heat-engine operation.
In order to exclude this kind of excitation, we apply artificial offset kicks to the ion. We
then invert the signal lines of both electrodes and observe that the phase of the resulting
oscillation is shifted by 180◦. If we apply only the electric noise to the electrodes, a swap
of the signal lines does not affect the phase of the oscillation. The ion is heated and its
probability distribution is increased, independent of the relative signs of the signals. The
result of this comparison is presented in Fig. 7.10. A phase shift of the oscillation under
an inversion of the signal lines indicates kicks due to an offset, while its absence shows
that a sign-independent energy source, namely the heating process, is responsible for
the oscillation. Consequently, Fig. 7.10 demonstrates that under heat-engine operation
conditions no additional offset or any other electrical kick contributes to the oscillation
of the ion. Thus, we can assign the total oscillatory energy to the heat engine drive and
the above evaluation of the engine parameters is justified.
118
7.8. Summary
7.8. Summary
In this chapter we have demonstrated a first realization of a heat engine whose working
agent is reduced to the limit of a single atom. We have shown that, indeed, a single
trapped ion is a well suited system to study single-particle thermodynamics. We have
driven the ion in a Stirling cycle, whereat the ion converted thermal and undirected
excitations into a coherent and directed motion. The versatility of the trapped ion system
allowed for the direct measurement of key features of the engine, such as thermodynamic
cycles, power output and the corresponding efficiency. These capacities are outstanding
characteristics of our system, allowing for a detailed analysis of the engine’s behavior.
While the efficiency is still low as compared to the Curzon-Ahlborn efficiency, it is
interesting to note that the relative output power scales in the same order of magnitude
as for macroscopic engines, although there is a difference of some dozen orders of
magnitude in the particle number. Thereby we showed that, as expected, the classical
framework of thermodynamics can be applied to our single-atom system, while the
notion of temperature requires a careful consideration. Our classical device offers a
broad platform for future experiments, investigating, for instance, thermal fluctuations
of work and heat, machines coupled to non-thermal reservoirs, see Sec. 3.3, nanoscopic
refrigerators and quantum heat engines, which will be discussed in Sec. 8.1 and Sec 8.3,
respectively.
119

8 Future perspectives of the single-
atom heat engine device
At the current state of the art of nanoscale engines and motors, the development of
machines for practical purposes seems to be reserved for molecular devices, which are
powered by light or electrochemical processes [Kay07]. Today, they are already produced
with high reliability and in large quantities. They are about to be assembled to larger
and more complex machines, completing diverse tasks such as transportation, rotation
or switching. Even cars on a molecular basis have been developed [Viv09]. However,
a future generation of those machines may operate in the quantum regime and on the
single-atom basis. Our single-atom engine is a pioneer towards this future generation of
single (few) atom devices.
After having presented the successful experimental realization of a single-atom heat
engine, we want to focus on the future perspectives of this experiment. Thereby, we will
present concrete approaches for the forthcoming investigations, as well as some rough
ideas where our findings may lead to.
The naturally following step is the inversion of our engine cycle in order to realize
a heat pump, acting as a single-atom refrigerator, which transfers heat between two
reservoirs at the expense of work. Two different setups for this device are discussed. Fur-
thermore, the heat engine operation may be refined by implementing an autonomously
running engine, where an external switching of the heat baths is not required anymore.
However, the trapped ion experiments are predestined for targeting the quantum regime.
To this end, we sketch different ideas of how to involve and to exploit quantum features
for future heat engine devices with trapped ions. We will end this outlook by a discussion
of applications of our setup beyond the purpose of thermodynamics and present our
system as a versatile tool for quantum computation.
8.1. Single-atom heat pump
In order to drive the ion in a heat pump operation, we essentially have to invert the
thermodynamic cycle of the engine [Kos00, Aba16]. A heat pump absorbs heat from
a cold reservoir and releases it to a hot reservoir, thereby increasing the temperature
difference between both. An external supply of work is required to drive the device,
since the heat pump is moving heat against the direction of spontaneous heat flow and
thus consumes energy [Lin10b, Kos14b].
121
8. Future perspectives of the single-atom heat engine device
D
Q
TH
A
C
TC
Q
B
Frequency ω
Figure 8.1. – Thermodynamic cycle of a heat pump for a harmonic oscillator (HO). The HO starts at
A with a hot bath temperature TH . Then, the frequency expands adiabatically and thereby the
temperature is reduced. At B, the HO temperature T is below the one of the cold bath, T < TC .
The HO is now coupled to the cold bath and thermalizes to T = TC at C. Next, the frequency
is compressed adiabatically, the temperature increases to T > TH at D. By thermalizing with
the hot bath, T = TH , the cycle is closed. As a result of this cycle, the HO absorbs heat from
the cold bath and releases it to the hot bath, at the expense of a net work consumption, and
therefore acts as a refrigerator.
Putting a nanoscopic heat pump to practical work is of high significance for future
applications. Heat production and, in particular, its dissipation is a crucial issue to nano
devices, since they provide little heat capacities and are supposed to operate isolated
from other systems [Cer09]. In future devices, smallest heat pumps may contribute to
the solution of this issue. In the follow we sketch two possible implementations of a heat
pump for our single trapped ion system.
8.1.1. Single-ion operation
For the single-ion system, the thermodynamic cycle may be inverted directly, maintaining
the same working principle as for the heat engine operation. To this end, the ion is driven
in a coherent oscillation by an external modulation of the trapping potential, see Sec. 5.1.1.
When the ion is oscillating, it is coupled to the hot and to the cold bath at the turning
points of its trajectory, but inverted as compared to the heat engine operation. This means
coupling to a hot bath when the ion is at the open end of the funnel and coupling to a cold
bath when the ion is at the narrow end. As a consequence, the thermodynamic cycle of
device is inverted. The ion absorbs work from the oscillation in order to move heat from
one side to the other. The resulting decrease of the oscillation amplitude corresponds to
the amount of energy consumed for the heat transport. However, the interaction with
the heat baths is emulated again, and therefore the baths themselves are not accessible to
measurements. In order to make the heat pump acting as a refrigerator, we need to make
sure that the virtual reservoir where we extract heat from has a lower temperature than
the reservoir where we release the heat to. Therefore, the expansion and compression
phases need to provide an energy change which is larger than the thermal energy change
during the reservoir interactions. The corresponding thermodynamic cycle is sketched
in Fig. 8.1.
122
Energy  H
8.1. Single-atom heat pump
a) b)
HEAT
Position z
Figure 8.2. – Three ion heat pump configuration. (a) The middle ion is performing the thermody-
namic cycle as shown in Fig. 8.1, while the outer ions play the role of the heat baths. When
driven in the "Egyptian mode", see arrows, the middle ion transfers heat from the ion with low
trap frequency (wide end of the tapered potential) to the ion at high trap frequency (narrow
end). (b) By the heat pump operation, a temperature gradient is generated in an initially
thermalized crystal, only by driving a coherent oscillation of the middle ion.
8.1.2. The three-ion heat pump
In order to access the temperatures of the reservoirs directly, a three ion approach is
favorable. Here, we assume a three ion Coulomb-crystal in linear configuration along
the trap axis, see Fig. 8.2. In this setup, the middle ion represents the heat pump,
while the outer ions play the role of reservoirs. We assume all ions to be initialized
at the same temperature. Now, we drive the "Egyptian mode" of the crystal, which
is the longitudinal mode with highest frequency, where the middle ion moves out of
phase with with twice the amplitude then the outer ions, see Fig. 8.2 (a) and [Kau12].
When the middle ion (heat pump) is moving along the trap axis, the trap frequency is
changed linearly. Thereby, the effective temperature changes linearly with ωz, according
to Eq. (6.2). We initially assume a thermalized crystal at T = T0. When the heat pump ion
moves from the narrow side of the funnel to the wide side, its state expands adiabatically.
When it reaches the reservoir ion, it provides a temperature T < T0. Now, the two
ions get in thermal contact and (partially) thermalize. Thereby, the outer ion is cooled
down and the heat pump ion is heated up. When the heat pump ion moves towards
the narrow end of the trap the inverse process occurs: its state is compressed and it
reaches the corresponding reservoir ion with T > T0. Consequently, this reservoir
ion is heated up when thermalizing with the heat pump ion. After some cycles, the
heat pump has generated a temperature difference between the two reservoir ions,
which have been initially at the same temperature. Note that the interaction between
radial modes of the ions is modified by the tapered trapping potential. Introducing
artificial dissipation in these modes allows to control the width and the overlap of the
resonances, and therefore enables to tune the coupling between the ions when they are
close together. A defined thermalization process allows for a conceptual separation
between the two reservoir interactions and compression/expansion phases. The final
temperature difference between the reservoir ions is proportional to the amplitude of the
oscillation of the heat pump ion.
123
Temperature T
8. Future perspectives of the single-atom heat engine device
a) b)
Figure 8.3. – Segmented trap design. (a) Model of a funnel-shaped ion trap with segmented elec-
trodes. Based on microfabricated chip traps in X-shape as in [Jac16]. (b) Possible shape of
the axial trapping potential (blue) in a segmented trap, in order to separate the cold reservoir
(blue ions), the hot reservoir (red ions) and the heat pump (gray ion).
In this configuration, it is beneficial to use a mixed crystal of different atomic species. If
the outer ions are much heavier than middle one, the relative amplitude of the middle ion
becomes larger and the outer ions may be assumed to remain stationary. Moreover, the
ions are resonant to different wavelengths; coupling optically to only one of the ions sim-
plifies the driving of the Egyptian mode drastically [Fel14]. As a consequence of expected
amplitudes in the range of 10 µm, stronger field gradients may be employed to increase
the resulting temperature difference, and more sensitive thermometry techniques, such
as sideband-spectroscopy, may allow to analyze the temperature difference.
Furthermore, using a trap with segmented electrodes allows for shaping the trapping
potential arbitrarily [Wal12b, Rus14]. Then, the heat pump ion and the reservoir ions
may be located in individual trapping potentials with individual trap frequencies, see
Fig. 8.3. This enables much larger oscillation amplitudes of the pumping ion, and avoids
an oscillation of the reservoir ions. In order to increase the heat capacity of the reservoirs,
larger ion crystals may be employed.
8.2. Autonomous, self-contained engine
One of the most illustrative picture of a self-contained machine is the Feynman ratchet,
which extracts work autonomously from an undirected Brownian motion [Rou94, Par02].
Also the famous drinking bird toy is a lively example of a self-contained, cyclic engine
process. Autonomous engines are particularly important for future implementations
of nanoscopic machines, which may be used for generation of work, energy harvesting
or refrigeration [Ton05, Woo16, All16]. They may find applications in nano sensors or
nanoscopic electronic devices, where an external switching of reservoir interactions is
not feasible.
Our single-ion engine may be operated in a self-driven manner in such a way, that the
oscillation of the system is used as feedback control for the bath couplings. This may be
achieved a natural way: when strongly focusing the radial cooling beam and pointing it
beside the ion but close to it, a strong gradient of the laser intensity is achieved along
the trap axis. Now, we can apply a global and continuous heating by electric field noise.
124
8.3. Towards a quantum heat engine
Figure 8.4. – Design for an autonomous operation of the single ion heat engine. (a) The cooling
laser beam (blue) is pointed slightly beside the rest position of the ion (green), while the ion
is heated continuously and globally by electric noise (red flashes). (b) This setup leads to a
spatial modulation of the effective temperature of the heat bath the ion is coupled to. As a
consequence, an initial oscillation of the ion is amplified in a self-contained manner.
The effective bath temperature is a superposition of the heating and cooling processes,
and for constant heating it linearly depends on the laser intensity, see Fig. 8.4. If the
ion is oscillating, its trajectory directly leads to a modulation of the heating and cooling
processes and the ion is driven in a Stirling-like engine cycle. As a consequence, the
heat engine principle amplifies an initial coherent oscillation, while the reservoirs are
provided in a continuous manner.
8.3. Towards a quantum heat engine
One of the major goals of single particle thermodynamics is to target the quantum regime
and to devise engines which are fueled by quantum effects or which generate and amplify
quantum states. The idea of a quantum heat engine is discussed extensively since the
late 1950s, when Scovil and Schulz-DuBois first showed that a three-level system may
act as a heat engine [Sco59], see Sec. 1.5. However, our single-ion system is the first
experimentally realized heat engine device, which has the potential to operate in the
quantum regime. By doing so, it would permit the study of the performance of small
quantum machines [Lin10b, Kos14b] and the exploration of genuine quantum effects in
thermodynamics, like quantum coherences [Scu03] and correlations [Dil09]. Note, that
in contrast to macroscopic and classical heat engines, on a quantum level the question
whether the Curzon-Ahlborn efficiency Eq. (2.9) is a universal bound on the efficiency at
maximum power is subject to ongoing discussions [Sch07, Esp09, Wan13a].
Having our system in mind, there are two different approaches to target the quantum
regime. On the one hand side, we may retain the working principles as discussed before,
but lowering the temperatures of the reservoirs close to the ground state of motion. On
the other side, we may not employ motional states but intrinsic quantum states to drive
the engine. Both ideas are sketched roughly in the following.
125
8. Future perspectives of the single-atom heat engine device
Figure 8.5. – Possible approaches to realize a quantum heat engine with a trapped-ion system. (a)
The temperature may be lowered in order to approach the ground state of motion, see text.
Again, we employ a funnel-shaped trap as before. The could bath may be at n̄ = 0, the hot
bath at, e.g. n̄ = 1. An Otto cycle may be implemented between the stages |a〉 = |ω1,n̄ = 0〉,
|b〉 = |ω2,0〉,|c〉 = |ω2,1〉 and |d〉 = |ω1,1〉. (b) Alternatively, the engine may employ the spin
of the trapped ion. Therefore, a spin dependent potential (black) is added to the trapping
potential (blue), converting a modulation in the occupation probability p of the spin states in
a coherent motion. p depends on the temperature of spin degree of freedom, see inset.
8.3.1. A heat engine at the ground state of motion
At low temperatures, kBT ≈ h̄ωr, the discreteness of the energy spectrum of the harmonic
oscillator plays an important role. Low temperatures may be reached by replacing
Doppler cooling by electromagnetically induced transparency cooling or side-band
cooling [Lei03], as in the recent verification of the quantum Jarzynski equality [An15].
The ion is then driven again in a thermodynamic cycle, employing a funnel-shaped
ion trap and the working principles as before, see Fig. 8.5 (a). In the quantum domain,
dark-state thermometry could be replaced by side-band spectroscopy. Therefore, the
ion has to be confined in strong trapping fields with trap frequencies high enough to
allow for resolved motional sidebands. This allows again for the determination of the
thermodynamic cycle of the engine and thus for a precise evaluation of its power and
efficiency.
8.3.2. The two-level system as heat engine
One of the intrinsic quantum states of an isolated atom is the spin of its electron shell,
leading to the fine structure splitting of the individual electron states. The 40Ca+ ion
provides a multiplicity of the ground state of 2, see Sec. 2.7, and may be employed as two-
level system. This is a paradigmatic system for analytical investigations and has been
analyzed in a variety of theoretical works [Gev92b, Fel00, Scu03, Kie04, Wan13a, Lon15a].
The engine may be driven either by quantum coherent excitations or by conventional
thermal excitations. In the latter case, a thermal excitation leads to an increasing pop-
ulation in the higher energy state, according to the Maxwell-Boltzmann distribution.
Having a perfectly spin polarized systems with all population in the lower state leads to
a minimal entropy and thus to T = 0. The highest temperature then corresponds to an
unpolarized state with probability p = 1/2 for both states. In order to generate work,
126
8.4. The tapered ion trap as a tool for quantum information
ωd
Position z
Figure 8.6. – Bow-tie configuration of a tapered ion trap for universal quantum computation op-
erations. (a) A linear ion crystal at the node of the bow-tie trap. (b) Expected radial trap
frequency ωr along the axis of the modified trap geometry. Ions located symmetrically around
the node of the bow-tie are excited by the same external driving frequency ωd. Within the
crystal, common modes mediate the excitation between the ions.
a change in the spin state must be translated into motion. This is naturally reached by
spin dependent forces, which may be realized by magnetic field gradients [Joh09] or by
optically induced ac-Stark shift gradients [Sch16].
Interestingly, occupation inversion in the two level system allows for the investigation
of negative temperatures [Bra13b]. This may reveal interesting effects on the dynamics of
heat engines, such as an efficiency of unity [Cam16]. However, the notion of negative tem-
peratures and their implications on engines systems is discussed controversially [Hil14].
Nonetheless, bath engineering techniques allow for an dissipative generation of di-
verse non-thermal quantum states to fuel the engine, employing coherences or quantum
correlations.
A converse approach is to employ the heat engine system to drive quantum states [Lev16].
Thereby, the work produced by the engine may be used to increase the energy of a quan-
tum state, such as a Fock state or an entangled state, as long as this state is generated in a
distinct degree of freedom and not destroyed by the reservoir interactions [GK14].
8.4. The tapered ion trap as a tool for quantum
information
Finally, we want to demonstrate the versatility of our novel trap design far beyond
the limits of thermodynamic applications. As already outlined in Sec. 5.4, the lifted
degeneracy of the radial trap frequencies, in combination with a tunable ion distance
and dissipation, promises to be helpful for the investigation of the transport of heat or
energy along ion chains.
Furthermore, the concept of a tapered trapping potential may provide a useful tool for
scalable quantum information processing. While outstanding quantum computational
operations have been performed with trapped ions in the last decade [Häf08, Bla12], a
major issue is still the scalability of the system [Mon13]. For universal computation oper-
ations, logic quantum gates have to be performed between arbitrary pairs of ions. One of
the most promising approaches is the division of larger crystals into smaller subsystems,
127
Radial trap frequency ωr
8. Future perspectives of the single-atom heat engine device
where logic gates are performed pairwise within these subsystems [Kie02]. Therefore,
reordering of the ion strings becomes necessary and complex shuttling, splitting and
swapping operations need to be performed [Wal12b, Rus14, Kau16].
However, using a double tapered trap in bow-tie geometry may facilitate these issues
dramatically. Therefore, we combine a funnel trap with a second identical, but inverted
trap, such that the potential is symmetric around the center, and a maximum of the trap
frequency is obtained in the symmetry plane, see Fig. 8.6. Now, for a linear ion crystal
at the center of the trap, always two ions in the left and in the right half of the crystal
provide the same trap frequency. Consequently, a common mode of the crystal exists
where only these two ions oscillate with a considerable amplitude. Therefore, logic gates
between these two ions may be performed deliberately. By moving the crystal around
the symmetry point, gates between any two ions may be performed. As a consequence,
such a trap design may boost the scalability of quantum computation devices.
128
A Appendix
A.1. Treatment of trap electrodes
As electrodes in the chosen trap design, Fig. 4.1, metallic rods are required which provide
a small outer diameter below 1mm and a length of at least 8mm. From a technical
point of view, it is challenging to trim electrodes from solid blocks or to use metal wires
such that they are perfectly straight over the length of trap and thus do not lead to
distortions of the trapping field. However, these requirements are found to be met
by commercial syringe-needles. They are made of hardened stainless steel with an
outer diameter of 0.8mm and provide a high manufacturing quality: They are perfectly
straight over a length of several centimeters and have a smooth surface on the micrometer
scale. An electron-microscope image of the surface of the syringe-needles is presented
in Fig. A.1 (a). However, the needles are produced with a light silicone coating, which
is advantageous for medical applications. But, such an insulating surface layer may
become electrostatically charged due to stray light from laser beams and thus may
disturb the trapping fields, leading to unwanted heating of trapped ions [Bro15]. The
effect of the silicone layer can be nicely identified on the electron microscope images
Fig. A.1 (a) & (b). We have recorded several images consecutively, exposing to increasing
larger sections. On each of the images, the section of the previous images is clearly visible
as square structure with different color shading. The electrons from previous images
have charged the silicone layer, which then reflects the following electrons differently
than the surrounding uncharged surface.
To be used as electrodes, the needles need a special surface treatment. To remove
the silicone layer we treated the needles with resolvents and with very fine sandpaper.
The resulting surface is shown in Fig. A.1 (c), revealing scratches on a micrometer scale,
but which are still small as compared to the diameter of the electrodes. However, the
charging effect during electron imaging has disappeared.
The needles are inserted in the appropriate holes in the ceramic endcap substrates. On
the outer surface, at the wide end side of the funnel, the needles are fixed with small
droplets of UHV compatible glue37. The electrical connection is made by self-made full
metal luster terminals.
37 Deltabond 152, Wakefield-Vette, Inc.
129
A. Appendix
Figure A.1. – Electron-microscope images of the syringe needles which are used as rf-electrodes.
(a) & (b) Needles before surface treatment. One recognizes the typical surface features of hot
rolled steel with a surface flatness on a µm scale. The rectangular structure on the surfaces in
the center of the images results from electron-charging from previous images. These features
point out the effect of surface charges on the silicone layer, which repel electrons during
subsequent image recording. (c) Needles after treatment. The traces from the sandpaper are
clearly visible on a scale of 10 µm, but the charging effect has disappeared.
A.2. Processing of PMT pulses
The photomultiplier tube (PMT) is the appropriate device to record the fluorescence of the
ion with short exposure times, as long as no spatial resolution is needed. For each photon,
the PMT generates a voltage pulse of 5V at a length of 5 ns. The count rate of the PMT is
limited by its dead time of about 100 ns, as a result of the changing process of its internal
dynodes. The PMT pulses are counted by a microcontroller (MC) unit, see Sec. 4.1.5.
However, the 5 ns pulses are too short to be recognized by the MC. To ensure a reliable
pulse counting, the 5 ns pulses are amplified and stretched to 50 ns, still shorter than the
dead time of the PMT. Therefore, we have designed an analogue circuit shown in Fig. A.2.
The pulse stretcher is based on a two stage transistor amplifier, where the lengthening
of the incoming pulses is based on the charging process of the capacitor C1, and thus
depends on the ratio of the capacity of C1 to the resistance R2. The modified voltage pulse
is fed to the MC. There, it fires an interrupt and thereby increases a counting variable.
While the PMT is running continuously, the exposure time is set by gating the MC. The
gating is triggered by the FPGA to guarantee precise timing with respect to the laser
pulses [Tol14]. The end of the gating pulse fires a second interrupt and the value of the
counter is written on the internal memory of the MC. In order to attain a high repetition
rate of the measurements, the integration of the PMT-counts can be done directly on the
internal memory of the MC with 128 kB. Thereby, we reach a maximum time resolution
of 2 µs and a maximum number of 2000 individual measurements. Subsequently, the
memory of MC is read and processed by the experiment control software, and the MC is
re-initialized [Tol14].
A.3. Stroboscopic recording of the ions motion
The ICCD camera allows to acquire images of the fluorescence with exposure times much
shorter than the trap frequency. This enables us to resolve a periodic motion of the ion.
Therefore, we drive the an oscillation of the ion and trigger the camera phase-locked to
the driving signal. An internal delay generator allows to scan the phase of the oscillation
where the image is taken. For each phase, a total exposure time of about 100ms is
130
A.4. Radio-frequency signal generator for AOM driving
5
5ns 50ns in
4 out
Q1 T1 3
BFR90 BF979
R4 2
1k 1
C1
0
20pF
0 100 200
Time (ns)
Figure A.2. – Circuit diagram for stretching the 5 ns pulses from the PMT to 50ns pulses, which
can be counted by the MC. We use an analogue transistor circuit, where the pulse extension
depends on the ratio of the capacity of C1 to the resistance R2. The measured incoming signal
from the PMT and the outgoing signal to the MC are shown on the right.
accumulated on the CCD chip, before the chip is read out. A time resolved oscillation is
demonstrated by the example of a breathing mode of a two-ion crystal, see Fig. A.3.
A.4. Radio-frequency signal generator for AOM driving
To drive the AOM we use a home built radio-frequency (rf) signal generator, which is
tunable in power an frequency by external control voltages. The output signal is turned
on and off by a TTL signal, which allows us to operate the AOMs as electro-optical
shutters and to switch the laser beams on and off. Furthermore, it allows us to adjust
the power of laser beams precisely and to tune their frequency within the range of the
according AOM. Thereby, the VCO device can be fully integrated into the electronic
experiment control system. A simplified block diagram of the circuit is shown in Fig. A.4.
The rf-signal is generated in by a VCO38, which is controlled either by an external or
by an internal control voltage. The external voltages are supplied by our stabilized
DAC-device, see Sec. 4.1.2. The internal control voltages are tuned by potentiometers. A
38VCO interchangable, we use POS-100, POS-150 and POS-300, Mini-Circuits.
Figure A.3. – Stroboscopic record of the breathing mode in axial direction of a two ion crystal at
141 kHz, acquired with the ICCD camera. Each image corresponds to a sum of 200000
exposures of 700 ns. The images are rotated by 90◦ (axial direction is vertical).
131
IN
R1
50
R2 +5V
330
R3 +5V
1k
OUT
Signal (V)
A. Appendix
+12V
26dB monitor
attenuator
0V
VCO switch mixer
frequency RF-output
modulation 14dB 18dB to AOM
0V...+10V +32dBminternal
external pre amp drive amp
+5V
sync-out
pulse
R
on
pulsed
external +12V
DC +12V
RF-enable opto
230V~ power
+5V
TTL 0V,5V coupler +15V
supply
0V -15V
amplitude 
modulation
0V...+6V internal
external
Figure A.4. – Block diagram of the controllable rf-signal generator which is used to drive the AOMs.
The signal is generated by a VCO. It can be switched and adjusted in power and frequency
by external control voltages. The rf-signal is amplified to a maximum level of 32 dBm, which
is enough to reach the maximum defraction efficiency of the AOM.
high frequency switch39 may block the transfer of the rf-signal to the AOM. The switch
is either turned steadily on, or controlled by an external TTL signal (decoupled by an
opto-coupler and passed via a transistor driving stage), or controlled by an internal
rectangular pulse generator40. The latter provides a duty cycle of 0.01 and reduces the
laser power behind the AOM accordingly. This mode of operation is used for a safe
beam adjustment at low optical power. A mixer multiplies the VCO signal by a control
voltage, which stems from an external or from the internal voltage source. A two stage
amplification (pre-amplifier41 and high power stage42) increases the output power to a
maximum level of 32 dBm, which is then fed to the AOM devices. Each VCO module
is provided with an separate 12V and 1A analogue power supply unit43 A monitor
signal allows to observe the VCO-frequency during operation. A characterization of the
output signal in terms of power and frequency as a function of the corresponding control
voltages is shown in Fig. A.5. We further characterized the switching behavior of the
AOMs, where we used the rf-enable TTL input of the VCO module. Fig. A.6 reveals a
switching of the laser power on a timescale of 50 ns. Each of the AOMs in our setup is
supplied by a n individual VCO module.
A.5. Power stabilization during AOM scans
The controllable VCO-module allows to perform frequency scans by tuning the AOM
frequency. As mentioned in Sec. 4.3.3, the internal electronic resonance circuit of the
39KSW-2-46+, Mini-Circuits.
40LMC555 CMOS Timer, Texas Instruments.
41ERA-5XSM+, Mini-Circuits.
421F5500B, RFHIC, not produced anymore. Replaced by RFC1G18H4-24, RFHIC. Needs an upgrade of the
supply voltage from 12V to 24V.
43HA15-0.9-A, CONDOR / SL Power Electronics Corp.
132
A.6. Switching between lasers with two VCOs
140
a) b)
1500
120
1000 100
80
500
60
0
40
0 1 2 3 4 5 6 7 −2 0 2 4 6 8 10 12 14 16
Control Voltage (V) Control Voltage (V)
Figure A.5. – Characteristics of the VCO module. (a) Power of the rf-output as a function of the
AM-control voltage. A voltage between 0V and 7V regulates the output power between
0mW and 1600mW, which corresponds to a maximum power level of 32 dBm. Measured
with a POS-100VCO (Mini-Circuits) at a frequency of 70MHz. (b) Frequency of the same VCO
as a function of the FM-control voltage. The frequency range between 40MHz to 130MHz is
reached by a control voltage between 0V and 15V. Above 10V, the maximum output power
of the VCO drops from 32dBm to 30dBm.
800
600
400
200
0
-200 -100 0 100 200
Time (ns)
Figure A.6. – Step-response of an AOM, operated as electro-optical shutter. The graph shows the
laser intensity, measured as a voltage from a photo-diode, as a function of time. After turning
off the rf-generator by an external TTL, the intensity of the laser beam decays within 50 ns.
AOMs is strongly frequency dependent. As a consequence, the defraction efficiency of
the AOM shows a dependance on the driving frequency, as measured in Fig. A.7 (a). To
avoid intensity fluctuations during frequency scans, the beam intensity is first measured
on a photo-diode (PD) as a function of the AOM drive frequency. For several grid points
of the driving frequency we correct the AOM driving power in order to obtain a constant
laser intensity. This results in a look-up table for corrected driving power values. These
tables are linearly interpolated and then used for subsequent frequency scans, leading to
a relative power stability of better than 99%.
A.6. Switching between lasers with two VCOs
In order to switch fast between two driving frequencies of the AOM, we employ two
individual VCO-modules. Both signal lines are combined by a rf-power combiner44 and
44ZSC-2-1W+, Mini-Circuits.
133
Voltage (mV) Output Power (mW)
Output Frequency (MHz)
A. Appendix
700 4.2 3200
a) b)
600 4
500
3.8 3000
400
3.6
300
3.4 2800
200
3.2
100
0 3 2600
−100 2.8
200 220 240 260 280 300 320 340 360 240 250 260 270 280 290 300 310
Drive Frequency (MHz) Drive Frequency (MHz)
Figure A.7. – Frequency dependance of the defraction efficiency of the AOM and its compensation.
(a) Optical power of the laser beam on a photo-diode (PD), after having passed the AOM in
double pass configuration. Here we characterize the TEF-270-100 AOM from Brimrose. (b) To
compensate for the non-uniform power distribution, we adjust the AM-control voltage for
each frequency (gray points) in order to reach a defined optical power (here the set point is
3000mV on the PD). Using the interpolated correction values, the standard deviation of the
optical power fluctuations during a scan of 50MHz, measured as a voltage on a photo-diode
(orange line), results to 29mV. The relative fluctuation is thus 29/3000 < 0.01.
0.2
0.1
0.0
-0.1
-0.2
0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22
Time HΜsL
Figure A.8. – Switching between two drive frequencies which for the AOM. Measured signal with
fitted sine-functions. The frequency is changed from 240MHz to 270MHz. In order to avoid
any photons with undesired frequencies in the transition regime, we turn off the driving
power during the frequency change for 50 ns. This is important for the pulsed temperature
measurement process, in order to avoid undesired heating or cooling effects by intermediate
frequencies.
134
Voltage PD (mV)
Amplitude rf-signal HmVL
Control Voltage (V)
Voltage PD (mV)
A.7. Design of a next generation funnel-shaped ion trap
1.30
a) b)
1.25
1.20
1.15
1.10
-150 -100 -50 0 50 100 150
Axial position (µm)
Figure A.9. –New design of the funnel-shaped ion trap. (a) Picture of the trap just before being
mounted in the vacuum chamber. More details regarding the geometry and the design are
presented in Fig. A.10. (b) Measured radial trap frequency as a function of the axial position.
Only the lower one of both radial frequencies has been recorded. The degeneracy has is lifted
by 5 kHz.
fed to the AOM. The FPGA-controlled TTLs enable the output of one module or the
other. This allows us to switch between two driving frequencies within 50 ns, see Fig. A.8.
This value is on the same timescale than the decay of the optical signal behind the
AOM, Fig. A.6, and much faster than the exposure time during the experiments, τexp ≤
5 µs. The switching between AOM frequencies is used during the pulsed temperature
measurements, see Fig. 6.8, where thermalization processes are resolved.
A.7. Design of a next generation funnel-shaped ion trap
We have developed a new generation of the funnel shaped ion trap, in order to face
the shortcomings of the old trap design, namely the low trap frequencies, the large size
of the trap, and a difficult mechanical alignment. We have implemented an X-shaped
blade-trap design, whose trapping volume is reduced by a factor of 16 as compared to the
rod trap used before. To achieve a high mechanical stability and an optimal alignment
of the electrodes, each pair of opposing electrodes is made from one cylindrical block
of stainless steel, see Fig. A.9 and Fig. A.10, similar to a fork with two prongs. Both
electrode pairs are then put together with an orientation of 90◦ to each other. The relative
alignment in all three dimensions is realized by four balls (diameter 3mm, grade 10),
made of silicon nitride Si3N4 (low thermal expansion, electrically insulating), which sit
in conical ball seats on both sides such that both metal parts are separated by 1mm.
The axis is defined by a common bore in solid metal holder. Concentric ceramic tubes,
fitted outside and inside the base cylinder of the electrode pairs ensure a high electric
insulation with a wall thickness of 3.5.mm and 2.5mm. The endcaps consist of stainless
steel cylinders, with pointed ends towards the trap volume. They are placed inside the
inner ceramic tube, having a mechanical stop to ensure a distance of 0.2mm between the
tips of the endcaps and the trap electrodes. The endcaps provide a concentric bore of
0.8mm to enable a laser beam to pass along the trap axis. Between the blade electrodes
we have placed compensation electrodes, using steel rods with a diameter of 2mm,
insulated within ceramic tubes.
The new trap has a length of 4mm and a radial distance between the trap axis and the
135
Radial trap frequency (MHz)
A. Appendix
a) b) c)
d)
e) f)
Figure A.10. – Composition of the new trap design. (a) One of the two fork-shaped electrode-pairs,
with its cylindrical base inside a ceramic insulator tube. Conical ball sits for the Al2O3 balls
are located on top of the fork prongs, as well as on the base. Additionally, a notch for the
access of a diagonal laser beam as well as holes for the compensation electrodes can be
seen. (b) The trap electrodes protrude from the fork, at the center of the image. The angle
of the electrodes provides 10◦. The second fork is identical, with an inverted orientation
of the tapered electrodes. (c) One of the endcaps, which are designed as elongated conical
rods. The edge in the upper part serves as mechanical stop to define a precise position of
the endcap. (d) Exploded view of the trap assembly. First the two fork-shaped electrode
pairs are fitted with 90◦ to each other by four silicon nitride balls (black). Then ceramic
tubes and the endcaps are inserted, each with a mechanical stop towards the trap volume.
From behind, the whole assembly is fixed with a metal clap, screwed to a solid mount. (e)
Quarter view of the trap design, including the holder and the compensation electrodes.
(f) Trap volume between the funnel shaped rf-electrodes and the endcaps after assembly.
Compensation electrodes are removed for a better view of the funnel electrodes.
136
A.7. Design of a next generation funnel-shaped ion trap
blade electrodes of 0.5mm at their narrow ending. The reduced size of the trap, as well
as a higher amplitude of the applied rf-voltage, enable for increased trap frequencies, see
Fig. A.9 (b). For radial trap frequencies around 1.2MHz, we measure a gradient of ωr(z)
along the trap axis in linear approximation of
ωr(z)/2π = (700(4)Hz/µm) · z+ 1187.8(3) kHz. (1.1)
The gradient is about a factor of 7 higher as compared to the old trap design, Eq. (7.1). By
the heat engine mechanism, the efficiency of the engine is supposed to increase linearly
with ∂ωr/∂z, allowing for a better performance and a higher output power. But, more
important, the higher trap frequencies allow for resolved sideband spectroscopy and
sideband cooling. Hence, cooling towards the ground state of motion is possible, which
enables for experiments in the quantum regime.
However, the blade-trap design provides an improved shielding of the trap volume
against external electric fields. While this fact is beneficial for precision experiments, it
makes a heating through electric noise more difficult: increased noise amplitudes by
about three orders of magnitude are needed. Alternatively, the ion may be heated using
a randomized photon scattering from a spectrally broadened laser beam. To avoid any
lock-in effect with the oscillation of the ion, the spectral width of the laser should be
larger than the Doppler broadened natural linewidth of the cooling transition of the ion.
137

B List of publications
The following articles have been published in the course of this thesis, where most parts
of the results have been presented:
A single-atom heat engine,
J. Roßnagel, S. Dawkins, K.N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler and
K. Singer,
Science 352, 325 (2016).
Experimental demonstration of a single-ion heat engine. We drive the ion thermally in
a Stirling cycle by coupling it alternately to hot and cold reservoirs. The output power
of the engine is used to drive a harmonic oscillation. From direct measurements of the
ion dynamics, we are able to determine the thermodynamic cycles and to evaluate the
power and efficiency of the engine.
Fast thermometry for trapped ions using dark resonances,
J. Roßnagel, K.N. Tolazzi, F. Schmidt-Kaler and K. Singer,
New J. of Phys. 17, 045004 (2015).
Demonstration of a method to determine the temperature of trapped ions, which is based
on the analysis of the lineshape of dark resonances in the fluorescence spectrum. The
method is accurate over a large dynamic range and allows for monitoring fast thermal-
ization processes, as required for analyzing the heat engine performance. Experimental
work is supported by a solution of the 8-level optical Bloch equations when including
the ions classical motion. Tuning the cooling laser to selected resonance features reveals
the capability to control and to adjust the ion’s temperature.
Nanoscale Heat Engine Beyond the Carnot Limit,
J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer and E. Lutz,
Phys. Rev. Lett. 112, 030602 (2014).
Editors’ Suggestion and featured in Physics.
Investigation of a single-ion heat engine driven by a squeezed thermal reservoir. We
show that the efficiency at maximum power increases with the degree of squeezing,
surpassing the standard Carnot limit for pure thermal excitations and approaching unity
exponentially for large squeezing parameters. We combine an analytical model for har-
monic oscillators with realistic numerical simulations of the system.
139
B. List of publications
Nano-Motor aus einem einzelnen Atom,
J. Roßnagel, K.N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler and K. Singer,
GIT Labor-Fachzeitschrift 6, 51 (2014).
Characterization measurements of the working principle of the heat engine and of the
tapered ion trap (in German).
Single-ion heat engine at maximum power,
O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer and E. Lutz,
Phys. Rev. Lett. 109, 203006 (2012).
Proposal for the single-ion heat-engine concept, containing an analytical framework
for the harmonic oscillator model system driven in a thermodynamic Otto cycle. We
demonstrate the feasibility of our setup by extensive numerical simulations, showing
that the engine is capable to operate at maximum output power with efficiencies of up to
30% under realistic conditions.
Publications not directly related to the selected main topic of this thesis:
Observation of the Kibble–Zurek scaling law for defect formation in ion crys-
tals,
S. Ulm, J. Roßnagel, G. Jacob, C. Degünther, S.T. Dawkins, U.G. Poschinger, R. Nig-
matullin, A. Retzker, M.B. Plenio, F. Schmidt-Kaler and K. Singer,
Nat. commun. 4, 2290 (2013).
Control of RILIS lasers at IGISOL facilities using a compact atomic beam ref-
erence cell,
T. Kron, R. Ferrer-Garcia, N. Lecesne, V. Sonnenschein, S. Raeder, J. Roßnagel,
K. Wendt,
Hyperfine Interact. 216, 53 (2013).
Determination of the first ionization potential of actinium,
J. Roßnagel, S. Raeder, A. Hakimi, R. Ferrer, N. Trautmann, and K. Wendt,
Phys. Rev. A 85, 012525 (2012).
Performance of a high repetition pulse rate laser system for in-gas-jet laser
ionization studies with the Leuven laser ion source @ LISOL,
R. Ferrer, V.T. Sonnenschein, B. Bastinc, J. Roßnagel et al.,
Nucl. Instr. Meth. Phys. Res. 291, 29 (2012).
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154
LEBENSLAUF  
 
PERSÖNLICHE DATEN 
Name Johannes Roßnagel 
Geboren am 27.08.1984 in Heidelberg 
Nationalität deutsch 
HOCHSCHULSTUDIUM 
01/2012 – 04/2017  Promotion in Physik 
Johannes Gutenberg-Universität Mainz  
Titel der Arbeit: A single atom heat engine 
bei Prof. Dr. Ferdinand Schmidt-Kaler 
10/2005 – 08/2011 Diplom Physik 
Johannes Gutenberg-Universität Mainz 
Diplomarbeit 
Titel der Arbeit: Aufbau einer Atomstrahl-
Massenspektrometer-Apparatur zur resonanten 
Laserionisation 
bei Prof. Dr. Klaus Wendt 
08/2007 – 06/2008 Auslandsstudium, Angewandte Physik 
 Université de Limoges, Frankreich 
Abschluss: Bachelor of Science (Licence de science) 
 
SCHULE 
05/2004 Abitur 
Kurfürst-Friedrich-Gymnasium, Heidelberg